Codebook design method for multiple input multiple output system and method for using the codebook

ABSTRACT

A multiple input multiple output (MIMO) communication method using a codebook is provided. The MIMO communication method may use one or more codebooks and the codebooks may change according to a transmission rank, a channel state of a user terminal, and/or a number of feedback bits.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application claims the benefit under 35 U.S.C. §119(e) of a U.S. Provisional Application No. 61/106,664, filed on Oct. 20, 2008, a U.S. Provisional Application No. 61/119,057, filed on Dec. 2, 2008, and a U.S. Provisional Application No. 61/141,441, filed on Dec. 30, 2008, in the United States Patent and Trademark Office, and the benefit under 35 U.S.C. §119(a) of a Korean Patent Application No. 10-2009-0081906, filed on Sep. 1, 2009, in the Korean Intellectual Property Office, the entire disclosures of which are incorporated herein by reference for all purposes.

BACKGROUND

1. Field

The following description relates to a design method of a codebook that is used in, for example, a multiple input multiple output (MIMO) communication system, and a technology to use the codebook.

2. Description of the Related Art

Researches are being conducted to provide various types of multimedia services and to support high quality and high speed of data transmission in a wireless communication environment. Technologies associated with a multiple input multiple output (MIMO) communication system using multiple channels are in rapid development.

In a MIMO communication system, a base station and terminals may use a codebook in order to securely and efficiently manage a channel environment. A particular space may be quantized into a plurality of codewords. The plurality of codewords that is generated by quantizing the particular space may be stored in the base station and the terminals. The codewords may be a vector or a matrix according to the dimension of a channel matrix.

For example, a terminal may select a matrix or a vector corresponding to channel information from matrices or vectors included in a codebook, based on a channel that is formed between the base station and the terminal. The base station may also receive the selected matrix or vector based on the codebook to thereby recognize the channel information. The selected matrix or vector may be used where the base station performs beamforming or transmits a transmission signal via multiple antennas.

SUMMARY

In one general aspect, provided herein is a multiple input multiple output (MIMO) communication system, the system comprising a terminal to feed back feedback data using a codebook, and a base station to access a memory storing the codebook, and to precode a data stream that the base station desires to transmit using the codebook.

Other features and aspects will be apparent from the following detailed description, the drawings, and the claims.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 illustrates an exemplary multiple input multiple output (MIMO) communication system.

FIG. 2 is a block diagram illustrating an exemplary configuration of a base station.

FIG. 3 is a flowchart illustrating an exemplary MIMO communication method.

FIG. 4 is a block diagram illustrating an exemplary base station and an exemplary terminal.

Throughout the drawings and the detailed description, unless otherwise described, the same drawing reference numerals will be understood to refer to the same elements, features, and structures. The relative size and depiction of these elements may be exaggerated for clarity, illustration, and convenience.

DETAILED DESCRIPTION

The following detailed description is provided to assist the reader in gaining a comprehensive understanding of the methods, apparatuses, and/or systems described herein. Accordingly, various changes, modifications, and equivalents of the systems, apparatuses, and/or methods described herein will be suggested to those of ordinary skill in the art. Also, description of well-known functions and constructions may be omitted for increased clarity and conciseness.

Hereinafter, exemplary embodiments will be described with reference to the accompanying drawings.

FIG. 1 illustrates an exemplary multiple input multiple output (MIMO) communication system. The MIMO communication system may be a closed-loop MIMO communication system or an open-loop MIMO communication system.

Referring to FIG. 1, the exemplary MIMO communication system includes a base station 110 and a plurality of users, for example, user 1, user 2, and user n_(u), represented by reference numerals 120, 130, and 140, respectively. While FIG. 1 shows an example of a multi-user MIMO communication system, it is understood that the disclosed systems, apparatuses, and/or methods may be applicable to a single user MIMO communication system. Herein, the term “closed-loop” may indicate that the users (user 1, user 2, user n_(u)) 120, 130, and 140 may feedback data to the base station 110. The feedback data may contain channel information and the base station 110 may generate a transmission signal based on the feedback data. The codebook described herein may be applicable to an open-loop MIMO communication system as well as the closed-loop MIMO communication system.

One or more antennas may be installed in the base station 110. A single antenna or a plurality of antennas may be installed in the users 120, 130, and/or 140. A channel may be formed between the base station 110 and the users 120, 130, and/or 140. Signals may be transmitted and received via each formed channel.

The base station 110 may transmit a single data stream or at least two data streams a user. For example, the base station 110 may adopt a spatial division multiplex access (SDMA) scheme or SDM scheme. The base station 110 may select a precoding matrix from codeword matrices included in a codebook and generate a transmission signal using the selected precoding matrix.

For example, the base station 110 may transmit pilot signals to the users 120, 130, and/or 140 via downlink channels. The pilot signals may be well known to the base station 110 and the users 120, 130, and/or 140.

A terminal corresponding to a user, for example, user 120, 130, or 140, may perform receiving a signal transmitted from the base station 110. The terminal may estimate a channel that is formed between the base station 110 and the terminal using a pilot signal. The terminal may select at least one matrix or vector from a codebook and feed back information associated with the selected at least one matrix or vector. The codebook may be updated according to a channel state. The codebook may be designed according to descriptions that will be made later with reference to FIGS. 2 through 4.

A channel estimator of the terminal may estimate the channel formed between the base station 110 and the terminal using the pilot signal. A terminal corresponding to a user, for example, user 120, 130, or 140 may select, as a vector, any one vector from vectors that are included in a pre-stored codebook. The terminal may select, as a codeword matrix, any one matrix from matrices that are included in the codebook.

For example, the terminal may select, as the vector or the codeword matrix, any one vector or any one matrix from 2^(B) vectors or 2^(B) matrices according to an achievable data transmission rate or a signal-to-interference and noise ratio (SINR). In this example, B denotes a number of feedback bits. A terminal may determine its own preferred transmission rank. The transmission rank may correspond to a number of data streams.

A feedback unit of the terminal may feedback to the base station 110, information associated with the selected vector or selected codeword matrix, hereinafter, referred to as channel information or feedback data. Information associated with the codeword matrix is also referred to as matrix information (PMI) or precoding matrix information (PMI). The channel information or the feedback data used herein may include channel state information, channel quality information, and/or channel direction information.

An information receiver of the base station 110 may receive channel information and/or feedback data of the users 120, 130, and/or 140, and determine a precoding matrix based on the received channel information and/or the feedback data. The base station 110 may select one or more of the users (user 1, user 2, user n_(u)) 120, 130, and 140 according to various types of selection algorithms, for example, a semi-orthogonal user selection (SUS) algorithm, a greedy user selection (GUS) algorithm, and the like.

The same codebook as the codebook that is stored in a memory the users 120, 130, and/or 140 may be pre-stored in a memory of the base station 110. The base station 110 may determine the precoding matrix based on matrices included in the pre-stored codebook using the channel information that is fed back from the users 120, 130, and/or 140. The base station 110 may determine the precoding matrix to maximize a total data transmission rate, such as a sum rate. The MIMO communication method described herein may achieve an efficient sum rate using an optimized codebook.

The base station 110 may precode data streams, for example, data streams S₁ and S_(N), based on the determined precoding matrix to generate a transmission signal. A process of generating the transmission signal by the base station 110 is referred to as “beamforming.” Beamforming is a signal processing technique used in sensor arrays for directional signal transmission or reception. This spatial selectivity is achieved by using adaptive or fixed receive/transmit beam patterns.

Generally, precoding is beamforming to support transmission in a radio system, for example, a multi-layer transmission in MIMO radio systems. Conventional beamforming considers linear single-layer precoding so that the same signal is emitted from each of the transmit antennas with appropriate weighting such that the signal power is maximized at the receiver output. When the receiver has multiple antennas, the single-layer beamforming cannot simultaneously maximize the signal level at all of the receive antennas and so precoding is used for multi-layer beamforming in order to maximize the throughput performance of a multiple receive antenna system. In precoding, the multiple streams of the signals are emitted from the transmit antennas with independent and appropriate weighting per each antenna such that the link throughput may be maximized at the receiver output. Precoding may be performed in a unitary fashion, in a semi-unitary fashion, or in a non-unitary fashion.

A channel environment between the base station 110 and the users 120, 130, and/or 140 may be variable. Where the base station 110 and the users 120, 130, and/or 140 use a fixed codebook, it may be difficult to adaptively cope with the varying channel environment. Although it will be described in detail later, the base station 110 and the users 120, 130, and/or 140 may adaptively cope with the varying channel environment to thereby update the codebook.

The base station 110 may generate a new precoding matrix using the updated codebook. For example, the base station 110 may update a previous precoding matrix to the new precoding matrix using the updated codebook.

FIG. 2 illustrates an exemplary configuration of a base station.

Referring to FIG. 2, the exemplary base station includes a layer mapping unit 210, a MIMO encoding unit 220, a precoder 230, and Z_(i) physical antennas 240.

At least one codeword for at least one user may be mapped to at least one layer. Where the dimension of codeword X is N_(C)×1, the layer mapping unit 210 may map the codeword X to the at least one layer using a matrix P with the dimension of N_(s)×N_(c). In this example, N_(s) denotes a number of layers or a number of effective antennas. Accordingly, it is possible to acquire the following Equation 1: s=Px  (1).

The MIMO encoding unit 220 may perform space-time modulation for S using a matrix function M with the dimension of N_(s)×N_(s). The MIMO encoding unit 220 may perform space-frequency block coding, spatial multiplexing, and the like, according to a transmission rank.

The precoder 230 may precode outputs, for example, output data streams of the MIMO encoding unit 220 to generate the transmission signal to be transmitted via the physical antennas 240. The dimension or the number of outputs, for example, the data streams of the MIMO encoding unit 220, may indicate the transmission rank. The precoder 230 may generate the transmission signal using a precoding matrix U with the dimension of N_(t)×N_(s). Accordingly, it is possible to acquire the following Equation 2: z=UM(s)  (2).

As described herein, W denotes the precoding matrix and R denotes the transmission rank or the number of effective antennas. In this example, the dimension of the precoding matrix W is N_(t)×R. Where the MIMO encoding unit 220 uses spatial multiplexing, Z may be given by the following Equation 3:

$\begin{matrix} {z = {{WB} = {{\begin{bmatrix} u_{11} & u_{1R} \\ \vdots & \vdots \\ u_{{Nt}\; 1} & u_{{Nt}\; R} \end{bmatrix}\begin{bmatrix} s_{1} \\ \vdots \\ s_{R} \end{bmatrix}}.}}} & (3) \end{matrix}$

Referring to the above Equation 3, the precoding matrix W may also be referred to as a weighting matrix. The dimension of the precoding matrix W may be determined according to the transmission rank and the number of physical antennas. For example, where the number N_(t) of physical antennas is four and the transmission rank is 2, the precoding matrix W may be given by the following Equation 4:

$\begin{matrix} {W = {\begin{bmatrix} W_{11} & W_{12} \\ W_{21} & W_{22} \\ W_{31} & W_{32} \\ W_{41} & W_{42} \end{bmatrix}.}} & (4) \end{matrix}$

Codebook Properties

The codebook used in a closed-loop MIMO communication system or an open-loop MIMO communication system may include one or more matrices and/or one or more vectors. A precoding matrix or a precoding vector may be determined based on the one or more matrices and/or the one or more vectors included in the codebook.

A first example of a codebook used in a single user MIMO communication system or a multi-user MIMO communication where the number of physical antennas of the base station is four, is shown in the following Equation 5:

$\begin{matrix} {{{W_{1} = {\frac{1}{\sqrt{2}}*U_{rot}*\begin{bmatrix} 1 & 1 & 0 & 0 \\ 1 & {- 1} & 0 & 0 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 1 & {- 1} \end{bmatrix}}},{W_{2} = {\frac{1}{\sqrt{2}}*U_{rot}*\begin{bmatrix} 1 & 1 & 0 & 0 \\ j & {- j} & 0 & 0 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & j & {- j} \end{bmatrix}}},{W_{3} = {\frac{1}{\sqrt{2}}*U_{rot}*\begin{bmatrix} 1 & 1 & 0 & 0 \\ 1 & {- 1} & 0 & 0 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & j & {- j} \end{bmatrix}}},{W_{4} = {\frac{1}{\sqrt{2}}*U_{rot}*\begin{bmatrix} 1 & 1 & 0 & 0 \\ j & {- j} & 0 & 0 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 1 & {- 1} \end{bmatrix}}},{W_{5} = {{{{diag}\left( {1,1,1,{- 1}} \right)}*{DFT}} = {0.5*\begin{bmatrix} 1 & 1 & 1 & 1 \\ 1 & j & {- 1} & {- j} \\ 1 & {- 1} & 1 & {- 1} \\ {- 1} & {j\;} & 1 & {- j} \end{bmatrix}}}},{and}}\begin{matrix} {W_{6} = {{{diag}\left( {1,\frac{\left( {1 + j} \right)}{\sqrt{2}},j\;,\frac{\left( {{- 1} + j} \right)}{\sqrt{2}}} \right)}*{DFT}}} \\ {= {0.5*{\begin{bmatrix} 1 & 1 & 1 & 1 \\ \frac{\left( {1 + j} \right)}{\sqrt{2}} & \frac{\left( {{- 1} + j} \right)}{\sqrt{2}} & \frac{\left( {{- 1} - j} \right)}{\sqrt{2}} & \frac{\left( {1 - j} \right)}{\sqrt{2}} \\ j & {- j} & j & {- j} \\ \frac{\left( {{- 1} + j} \right)}{\sqrt{2}} & \frac{\left( {1 + j} \right)}{\sqrt{2}} & \frac{\left( {1 - j} \right)}{\sqrt{2}} & \frac{\left( {{- 1} - j} \right)}{\sqrt{2}} \end{bmatrix}.}}} \end{matrix}} & (5) \end{matrix}$

In this example, a rotation matrix is

$\begin{matrix} {{U_{{rot}\;} = {\frac{1}{\sqrt{2}}\begin{bmatrix} 1 & 0 & {- 1} & 0 \\ 0 & 1 & 0 & {- 1} \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \end{bmatrix}}},} & \; \end{matrix}$ and a quadrature phase shift keying (QPSK) discrete Fourier transform (DFT) matrix is

${{DFT} = {0.5*\begin{bmatrix} 1 & 1 & 1 & 1 \\ 1 & j & {- 1} & {- j} \\ 1 & {- 1} & 1 & {- 1} \\ 1 & {- j} & {- 1} & j \end{bmatrix}}},$ diag(a, b, c, d) is a 4×4 matrix, and diagonal elements of diag(a, b, c, d) are a, b, c, and d, and all the remaining elements are zero.

16 matrices included in the 4-bit codebook for the single user MIMO communication system may be determined according to the transmission rank, as given by the following Table 1:

Transmit Codebook Transmission Transmission Transmission Transmission Index Rank 1 Rank 2 Rank 3 Rank 4 1 C_(1,1) = W1(;,2) C_(1,2) = W1(;,1 2) C_(1,3) = W1(;,1 2 3) C_(1,4) = W1(;,1 2 3 4) 2 C_(2,1) = W1(;,3) C_(2,2) = W1(;,1 3) C_(2,3) = W1(;,1 2 4) C_(2,4) = W2(;,1 2 3 4) 3 C_(3,1) = W1(;,4) C_(3,2) = W1(;,1 4) C_(3,3) = W1(;,1 3 4) C_(3,4) = W3(;,1 2 3 4) 4 C_(4,1) = W2(;,2) C_(4,2) = W1(;,2 3) C_(4,3) = W1(;,2 3 4) C_(4,4) = W4(;,1 2 3 4) 5 C_(5,1) = W2(;,3) C_(5,2) = W1(;,2 4) C_(5,3) = W2(;,1 2 3) C_(5,4) = W5(;,1 2 3 4) 6 C_(6,1) = W2(;,4) C_(6,2) = W1(;,3 4) C_(6,3) = W2(;,1 2 4) C_(6,4) = W6(;,1 2 3 4) 7 C_(7,1) = W3(;,1) C_(7,2) = W2(;,1 3) C_(7,3) = W2(;,1 3 4) n/a 8 C_(8,1) = W4(;,1) C_(8,2) = W2(;,1 4) C_(8,3) = W2(;,2 3 4) n/a 9 C_(9,1) = W5(;,1) C_(9,2) = W2(;,2 3) C_(9,3) = W3(;,1 2 3) n/a 10 C_(10,1) = W5(;,2) C_(10,2) = W2(;,2 4) C_(10,3) = W3(;,1 3 4) n/a 11 C_(11,1) = W5(;,3) C_(11,2) = W3(;,1 3) C_(11,3) = W4(;,1 2 3) n/a 12 C_(12,1) = W5(;,4) C_(12,2) = W3(;,1 4) C_(12,3) = W4(;,1 3 4) n/a 13 C_(13,1) = W6(;,1) C_(13,2) = W4(;,1 3) C_(13,3) = W5(;,1 2 3) n/a 14 C_(14,1) = W6(;,2) C_(14,2) = W4(;,1 4) C_(14,3) = W5(;,1 3 4) n/a 15 C_(15,1) = W6(;,3) C_(15,2) = W5(;,1 3) C_(15,3) = W6(;,1 2 4) n/a 16 C_(16,1) = W6(;,4) C_(16,2) = W6(;,2 4) C_(16,3) = W6(;,2 3 4) n/a

Referring to the above Table 1, where the transmission rank is 4, the precoding matrix may be generated based on, for example, W₁(;,1 2 3 4), W₂(;,1 2 3 4), W₃(;,1 2 3 4), W₄(;,1 2 3 4), W₅(;,1 2 3 4), and W₆(;,1 2 3 4). In this example, W_(k)(;,n m o p) denotes a matrix that includes an n^(th) column vector, an m^(th) column vector, an o^(th) column vector, and a p^(th) column vector of W_(k).

Where the transmission rank is 3, the precoding matrix may be generated based on, for example, W₁(;,1 2 3), W₁(;,1 2 4), W₁(;,1 3 4), W₁(;,2 3 4), W₂(;,1 2 3), W₂(;,1 2 4), W₂(;,1 3 4), W₂(;,2 3 4), W₃(;,1 2 3), W₃(;,1 3 4), W₄(;,1 2 3), W₄(;,1 3 4), W₅(;,1 2 3), W₅(;,1 3 4), W₆(;,1 2 4), and W₆(;,2 3 4). In this example, W_(k)(;,n m o) denotes a matrix that includes the n^(th) column vector, the m^(th) column vector, and the o^(th) column vector of W_(k).

Where the transmission rank is 2, the precoding matrix may be generated based on, for example, W₁(;,1 2), W₁(;,1 3), W₁(;,1 4), W₁(;,2 3), W₁(;,2 4), W₁(;,3 4), W₂(;,1 3), W₂(;,1 4), W₂(;,2 3), W₂(;,2 4), W₃(;,1 3), W₃(;,1 4), W₄(;,1 3), W₄(;,1 4), W₅(;,1 3), and W₆(;,2 4). In this example, W_(k)(;,n m) denotes a matrix that includes the n^(th) column vector and the m^(th) column vector of W_(k).

Where the transmission rank is 1, the precoding matrix may be generated based on, for example, W₁(;,2), W₁(;,3), W₁(;,4), W₂(;,2), W₂(;,3), W₂(;,4), W₃(;,1), W₄(;,1), W₅(;,1), W₅(;,2), W₅(;,3), W₅(;,4), W₆(;,1), W₆(;,2), W₆(;,3), and W₆(;,4). In this example, W_(k)(;,n) denotes the n^(th) column vector of W_(k).

The codewords included in the above Table 1 may be expressed equivalently as follows:

$C_{1,1} = \begin{matrix} 0.5000 \\ {- 0.5000} \\ 0.5000 \\ {- 0.5000} \end{matrix}$ $C_{2,1} = \begin{matrix} {- 0.5000} \\ {- 0.5000} \\ 0.5000 \\ 0.5000 \end{matrix}$ $C_{3,1} = \begin{matrix} {- 0.5000} \\ 0.5000 \\ 0.5000 \\ {- 0.5000} \end{matrix}$ $C_{4,1} = \begin{matrix} 0.5000 \\ {0 - {0.5000{\mathbb{i}}}} \\ 0.5000 \\ {0 - {0.5000{\mathbb{i}}}} \end{matrix}$

$C_{5,1} = \begin{matrix} {- 0.5000} \\ {0 - {0.5000{\mathbb{i}}}} \\ 0.5000 \\ {0 + {0.5000{\mathbb{i}}}} \end{matrix}$ $C_{6,1} = \begin{matrix} {- 0.5000} \\ {0 + {0.5000{\mathbb{i}}}} \\ 0.5000 \\ {0 - {0.5000{\mathbb{i}}}} \end{matrix}$ $C_{7,1} = \begin{matrix} 0.5000 \\ 0.5000 \\ 0.5000 \\ 0.5000 \end{matrix}$ $C_{8,1} = \begin{matrix} 0.5000 \\ {0 + {0.5000{\mathbb{i}}}} \\ 0.5000 \\ {0 + {0.5000{\mathbb{i}}}} \end{matrix}$

$C_{9,1} = \begin{matrix} 0.5000 \\ 0.5000 \\ 0.5000 \\ {- 0.5000} \end{matrix}$ $C_{10,1} = \begin{matrix} 0.5000 \\ {0 + {0.5000{\mathbb{i}}}} \\ {- 0.5000} \\ {0 + {0.5000{\mathbb{i}}}} \end{matrix}$ $C_{11,1} = \begin{matrix} 0.5000 \\ {- 0.5000} \\ 0.5000 \\ 0.5000 \end{matrix}$ $C_{12,1} = \begin{matrix} 0.5000 \\ {0 - {0.5000{\mathbb{i}}}} \\ {- 0.5000} \\ {0 - {0.5000{\mathbb{i}}}} \end{matrix}$

$C_{13,1} = \begin{matrix} 0.5000 \\ {03536 + {0.3536{\mathbb{i}}}} \\ {0 + {0.5000{\mathbb{i}}}} \\ {{- 0.3536} + {0.3536{\mathbb{i}}}} \end{matrix}$ $C_{14,1} = \begin{matrix} 0.5000 \\ {{- 03536} + {0.3536{\mathbb{i}}}} \\ {0 - {0.5000{\mathbb{i}}}} \\ {0.3536 + {0.3536{\mathbb{i}}}} \end{matrix}$ $C_{15,1} = \begin{matrix} 0.5000 \\ {{- 03536} - {0.3536{\mathbb{i}}}} \\ {0 + {0.5000{\mathbb{i}}}} \\ {0.3536 - {0.3536{\mathbb{i}}}} \end{matrix}$ $C_{16,1} = \begin{matrix} 0.5000 \\ {03536 - {0.3536{\mathbb{i}}}} \\ {0 - {0.5000{\mathbb{i}}}} \\ {{- 0.3536} - {0.3536{\mathbb{i}}}} \end{matrix}$

$C_{1,2} = \begin{matrix} 0.5000 & 0.5000 \\ 0.5000 & {- 0.5000} \\ 0.5000 & 0.5000 \\ 0.5000 & {- 0.5000} \end{matrix}$ $C_{2,2} = \begin{matrix} 0.5000 & {- 0.5000} \\ 0.5000 & {- 0.5000} \\ 0.5000 & 0.5000 \\ 0.5000 & 0.5000 \end{matrix}$ $C_{3,2} = \begin{matrix} 0.5000 & {- 0.5000} \\ 0.5000 & 0.5000 \\ 0.5000 & 0.5000 \\ 0.5000 & {- 0.5000} \end{matrix}$ $C_{4,2} = \begin{matrix} 0.5000 & {- 0.5000} \\ {- 0.5000} & {- 0.5000} \\ 0.5000 & 0.5000 \\ {- 0.5000} & 0.5000 \end{matrix}$

$C_{5,2} = \begin{matrix} 0.5000 & {- 0.5000} \\ {- 0.5000} & 0.5000 \\ 0.5000 & 0.5000 \\ {- 0.5000} & {- 0.5000} \end{matrix}$ $C_{6,2} = \begin{matrix} {- 0.5000} & {- 0.5000} \\ {- 0.5000} & 0.5000 \\ 0.5000 & 0.5000 \\ 0.5000 & {- 0.5000} \end{matrix}$ $C_{7,2} = \begin{matrix} 0.5000 & {- 0.5000} \\ {0 + {0.5000{\mathbb{i}}}} & {0 - {0.5000\;{\mathbb{i}}}} \\ 0.5000 & 0.5000 \\ {0 + {0.5000{\mathbb{i}}}} & {0 + {0.5000{\mathbb{i}}}} \end{matrix}$ $C_{8,2} = \begin{matrix} 0.5000 & {- 0.5000} \\ {0 + {0.5000{\mathbb{i}}}} & {0 + {0.5000{\mathbb{i}}}} \\ 0.5000 & 0.5000 \\ {0 + {0.5000{\mathbb{i}}}} & {0 - {0.5000\;{\mathbb{i}}}} \end{matrix}$

$C_{9,2} = \begin{matrix} 0.5000 & {- 0.5000} \\ {0 - {0.5000{\mathbb{i}}}} & {0 - {0.5000\;{\mathbb{i}}}} \\ 0.5000 & 0.5000 \\ {0 - {0.5000{\mathbb{i}}}} & {0 + {0.5000\;{\mathbb{i}}}} \end{matrix}$ $C_{10,2} = \begin{matrix} 0.5000 & {- 0.5000} \\ {0 - {0.5000{\mathbb{i}}}} & {0 + {0.5000\;{\mathbb{i}}}} \\ 0.5000 & 0.5000 \\ {0 - {0.5000{\mathbb{i}}}} & {0 - {0.5000{\mathbb{i}}}} \end{matrix}$ $C_{11,2} = \begin{matrix} 0.5000 & {- 0.5000} \\ 0.5000 & {0 - {0.5000\;{\mathbb{i}}}} \\ 0.5000 & 0.5000 \\ 0.5000 & {0 + {0.5000\;{\mathbb{i}}}} \end{matrix}$ $C_{12,2} = \begin{matrix} 0.5000 & {- 0.5000} \\ 0.5000 & {0 + {0.5000{\mathbb{i}}}} \\ 0.5000 & 0.5000 \\ 0.5000 & {0 - {0.5000\;{\mathbb{i}}}} \end{matrix}$

$C_{13,2} = \begin{matrix} 0.5000 & {- 0.5000} \\ {0 + {0.5000{\mathbb{i}}}} & {- 0.5000} \\ 0.5000 & 0.5000 \\ {0 + {0.5000{\mathbb{i}}}} & 0.5000 \end{matrix}$ $C_{14,2} = \begin{matrix} 0.5000 & {- 0.5000} \\ {0 + {0.5000{\mathbb{i}}}} & 0.5000 \\ 0.5000 & 0.5000 \\ {0 + {0.5000{\mathbb{i}}}} & {- 0.5000} \end{matrix}$ $C_{15,2} = \begin{matrix} 0.5000 & 0.5000 \\ 0.5000 & {- 0.5000} \\ 0.5000 & 0.5000 \\ {- 0.5000} & 0.5000 \end{matrix}$ $C_{16,2} = \begin{matrix} 0.5000 & 0.5000 \\ {{- 0.3536} + {0.3536{\mathbb{i}}}} & {0.3536 - {0.3536{\mathbb{i}}}} \\ {0 - {0.5000{\mathbb{i}}}} & {0 - {0.5000{\mathbb{i}}}} \\ {0.3536 + {0.3536{\mathbb{i}}}} & {{- 0.3536} - {0.3536{\mathbb{i}}}} \end{matrix}$

$C_{1,3} = \begin{matrix} 0.5000 & 0.5000 & {- 0.5000} \\ 0.5000 & {- 0.5000} & {- 0.5000} \\ 0.5000 & 0.5000 & 0.5000 \\ 0.5000 & {- 0.5000} & 0.5000 \end{matrix}$ $C_{2,3} = \begin{matrix} 0.5000 & 0.5000 & {- 0.5000} \\ 0.5000 & {- 0.5000} & 0.5000 \\ 0.5000 & 0.5000 & 0.5000 \\ 0.5000 & {- 0.5000} & {- 0.5000} \end{matrix}$ $C_{3,3} = \begin{matrix} 0.5000 & {- 0.5000} & {- 0.5000} \\ 0.5000 & {- 0.5000} & 0.5000 \\ 0.5000 & 0.5000 & 0.5000 \\ 0.5000 & 0.5000 & {- 0.5000} \end{matrix}$ $C_{4,3} = \begin{matrix} 0.5000 & {- 0.5000} & {- 0.5000} \\ {- 0.5000} & {- 0.5000} & 0.5000 \\ 0.5000 & 0.5000 & 0.5000 \\ {- 0.5000} & 0.5000 & {- 0.5000} \end{matrix}$

$C_{5,3} = \begin{matrix} 0.5000 & 0.5000 & {- 0.5000} \\ {{0 + {0.5000\;{\mathbb{i}}}}\;} & {0 - {0.5000{\mathbb{i}}}} & {0 - {0.5000\;{\mathbb{i}}}} \\ 0.5000 & 0.5000 & 0.5000 \\ {0 + {0.5000\;{\mathbb{i}}}} & {0 - {0.5000\;{\mathbb{i}}}} & {0 + {0.5000\;{\mathbb{i}}}} \end{matrix}$ $C_{6,3} = \begin{matrix} 0.5000 & 0.5000 & {- 0.5000} \\ {0 + {0.5000\;{\mathbb{i}}}} & {0 - {0.5000\;{\mathbb{i}}}} & {0 + {0.5000\;{\mathbb{i}}}} \\ 0.5000 & 0.5000 & 0.5000 \\ {0 + {0.5000\;{\mathbb{i}}}} & {0 - {0.5000\;{\mathbb{i}}}} & {0 - {0.5000\;{\mathbb{i}}}} \end{matrix}$ $C_{7,3} = \begin{matrix} 0.5000 & {- 0.5000} & {- 0.5000} \\ {0 + {0.5000\;{\mathbb{i}}}} & {0 - {0.5000\;{\mathbb{i}}}} & {0 + {0.5000\;{\mathbb{i}}}} \\ 0.5000 & 0.5000 & 0.5000 \\ {0 + {0.5000\;{\mathbb{i}}}} & {0 + {0.5000\;{\mathbb{i}}}} & {0 - {0.5000\;{\mathbb{i}}}} \end{matrix}$ $C_{8,3} = \begin{matrix} 0.5000 & {- 0.5000} & {- 0.5000} \\ {0 - {0.5000\;{\mathbb{i}}}} & {0 - {0.5000\;{\mathbb{i}}}} & {0 + {0.5000\;{\mathbb{i}}}} \\ 0.5000 & 0.5000 & 0.5000 \\ {0 - {0.5000\;{\mathbb{i}}}} & {0 + {0.5000\;{\mathbb{i}}}} & {0 - {0.5000\;{\mathbb{i}}}} \end{matrix}$

$C_{9,3} = \begin{matrix} 0.5000 & 0.5000 & {- 0.5000} \\ 0.5000 & {- 0.5000} & {0 - {0.5000{\mathbb{i}}}} \\ 0.5000 & 0.5000 & 0.5000 \\ 0.5000 & {- 0.5000} & {0 + {0.5000{\mathbb{i}}}} \end{matrix}$ $C_{10,3} = \begin{matrix} 0.5000 & {- 0.5000} & {- 0.5000} \\ 0.5000 & {0 - {0.5000\;{\mathbb{i}}}} & {0 + {0.5000\;{\mathbb{i}}}} \\ 0.5000 & 0.5000 & 0.5000 \\ 0.5000 & {0 + {0.5000\;{\mathbb{i}}}} & {0 - {0.5000\;{\mathbb{i}}}} \end{matrix}$ $C_{11,3} = \begin{matrix} 0.5000 & 0.5000 & {- 0.5000} \\ {0 + {0.5000{\mathbb{i}}}} & {0 - {0.5000\;{\mathbb{i}}}} & {- 0.5000} \\ 0.5000 & 0.5000 & 0.5000 \\ {0 + {0.5000{\mathbb{i}}}} & {0 - {0.5000\;{\mathbb{i}}}} & 0.5000 \end{matrix}$ $C_{12,3} = \begin{matrix} 0.5000 & {- 0.5000} & {- 0.5000} \\ {0 + {0.5000{\mathbb{i}}}} & {- 0.5000} & 0.5000 \\ 0.5000 & 0.5000 & 0.5000 \\ {0 + {0.5000{\mathbb{i}}}} & 0.5000 & {- 0.5000} \end{matrix}$

$C_{13,3} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 \\ 0.5000 & {0 + {0.5000\;{\mathbb{i}}}} & {- 0.5000} \\ 0.5000 & {- 0.5000} & 0.5000 \\ {- 0.5000} & {0 + {0.5000\;{\mathbb{i}}}} & 0.5000 \end{matrix}$ $C_{14,3} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 \\ 0.5000 & {- 0.5000} & {0 - {0.5000\;{\mathbb{i}}}} \\ 0.5000 & 0.5000 & {- 0.5000} \\ {- 0.5000} & 0.5000 & {0 - {0.5000\;{\mathbb{i}}}} \end{matrix}$ $C_{15,3} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 \\ {0.3536 + {0.3536{\mathbb{i}}}} & {{- 0.3536} + {0.3536{\mathbb{i}}}} & {0.3536 - {0.3536{\mathbb{i}}}} \\ {0 + {0.5000{\mathbb{i}}}} & {0 - {0.5000{\mathbb{i}}}} & {0 - {0.5000{\mathbb{i}}}} \\ {{- 0.3536} + {0.3536{\mathbb{i}}}} & {0.3536 + {0.3536{\mathbb{i}}}} & {{- 0.3536} - {0.3536{\mathbb{i}}}} \end{matrix}$ $C_{16,3} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 \\ {{- 0.3536} + {0.3536{\mathbb{i}}}} & {{- 0.3536} - {0.3536{\mathbb{i}}}} & {0.3536 - {0.3536{\mathbb{i}}}} \\ {0 - {0.5000{\mathbb{i}}}} & {0 + {0.5000{\mathbb{i}}}} & {0 - {0.5000{\mathbb{i}}}} \\ {0.3536 + {0.3536{\mathbb{i}}}} & {0.3536 - {0.3536{\mathbb{i}}}} & {{- 0.3536} - {0.3536{\mathbb{i}}}} \end{matrix}$

$C_{1,4} = \begin{matrix} 0.5000 & 0.5000 & {- 0.5000} & {- 0.5000} \\ 0.5000 & {- 0.5000} & {- 0.5000} & 0.5000 \\ 0.5000 & 0.5000 & 0.5000 & 0.5000 \\ 0.5000 & {- 0.5000} & 0.5000 & {- 0.5000} \end{matrix}$ $C_{2,4} = \begin{matrix} 0.5000 & 0.5000 & {- 0.5000} & {- 0.5000} \\ {0 + {0.5000{\mathbb{i}}}} & {0 - {0.5000{\mathbb{i}}}} & {0 - {0.5000{\mathbb{i}}}} & {0 + {0.5000{\mathbb{i}}}} \\ 0.5000 & 0.5000 & 0.5000 & 0.5000 \\ {0 + {0.5000{\mathbb{i}}}} & {0 - {0.5000{\mathbb{i}}}} & {0 + {0.5000{\mathbb{i}}}} & {0 - {0.5000{\mathbb{i}}}} \end{matrix}$ $C_{3,4} = \begin{matrix} 0.5000 & 0.5000 & {- 0.5000} & {- 0.5000} \\ 0.5000 & {- 0.5000} & {0 - {0.5000{\mathbb{i}}}} & {0 + {0.5000{\mathbb{i}}}} \\ 0.5000 & 0.5000 & 0.5000 & 0.5000 \\ 0.5000 & {- 0.5000} & {0 + {0.5000{\mathbb{i}}}} & {0 - {0.5000{\mathbb{i}}}} \end{matrix}$ $C_{4,4} = \begin{matrix} 0.5000 & 0.5000 & {- 0.5000} & {- 0.5000} \\ {0 + {0.5000{\mathbb{i}}}} & {0 - {0.5000{\mathbb{i}}}} & {- 0.5000} & 0.5000 \\ 0.5000 & 0.5000 & 0.5000 & 0.5000 \\ {0 + {0.5000{\mathbb{i}}}} & {0 - {0.5000{\mathbb{i}}}} & 0.5000 & {- 0.5000} \end{matrix}$

$C_{5,4} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 & 0.5000 \\ 0.5000 & {0 + {0.5000{\mathbb{i}}}} & {- 0.5000} & {0 - {0.5000{\mathbb{i}}}} \\ 0.5000 & {- 0.5000} & 0.5000 & {- 0.5000} \\ {- 0.5000} & {0 + {0.5000{\mathbb{i}}}} & 0.5000 & {0 - {0.5000{\mathbb{i}}}} \end{matrix}$ $C_{6,4} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 & 0.5000 \\ {0.3536 + {0.3536{\mathbb{i}}}} & {{- 0.3536} + {0.3536{\mathbb{i}}}} & {{- 0.3536} - {0.3536{\mathbb{i}}}} & {0.3536 - {0.3536{\mathbb{i}}}} \\ {0 + {0.5000{\mathbb{i}}}} & {0 - {0.5000{\mathbb{i}}}} & {0 + {0.5000{\mathbb{i}}}} & {0 - {0.5000{\mathbb{i}}}} \\ {{- 0.3536} + {0.3536{\mathbb{i}}}} & {0.3536 + {0.3536{\mathbb{i}}}} & {0.3536 - {0.3536{\mathbb{i}}}} & {{- 0.3536} - {0.3536{\mathbb{i}}}} \end{matrix}$

For the first example, where the number of physical antennas of the base station is four, the codebook used in the multi-user MIMO communication system according to an exemplary embodiment may be designed by the following Equation 6:

$\begin{matrix} {{W_{3} = {\frac{1}{\sqrt{2}}*U_{rot}*\begin{bmatrix} 1 & 1 & 0 & 0 \\ 1 & {- 1} & 0 & 0 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & j & {- j} \end{bmatrix}}}\begin{matrix} {W_{6} = {{{diag}\left( {1,\frac{\left( {1 + j} \right)}{\sqrt{2}},j,\frac{\left( {{- 1} + j} \right)}{\sqrt{2}}} \right)}*{DFT}}} \\ {= {0.5*{\begin{bmatrix} 1 & 1 & 1 & 1 \\ \frac{\left( {1 + j} \right)}{\sqrt{2}} & \frac{\left( {{- 1} + j} \right)}{\sqrt{2}} & \frac{\left( {{- 1} - j} \right)}{\sqrt{2}} & \frac{\left( {1 - j} \right)}{\sqrt{2}} \\ j & {- j} & j & {- j} \\ \frac{\left( {{- 1} + j} \right)}{\sqrt{2}} & \frac{\left( {1 + j} \right)}{\sqrt{2}} & \frac{\left( {1 - j} \right)}{\sqrt{2}} & \frac{\left( {{- 1} - j} \right)}{\sqrt{2}} \end{bmatrix}.}}} \end{matrix}} & (6) \end{matrix}$

Codewords included in the codebook for the multi-user MIMO communication system may be expressed using a transmission rank as given by the following Table 2:

codeword used at Transmit Codebook the mobile station Index for quantization 1 M₁=W3(;,1) 2 M₂=W3(;,2) 3 M₃=W3(;,3) 4 M₄=W3(;,4) 5 M₅=W6(;,1) 6 M₆=W6(;,2) 7 M₇=W6(;,3) 8 M₈=W6(;,4)

Referring to the above Table 2, where a transmission rank of each of users is one, the precoding matrix may be constructed by combining W₃(;1), W₃(;2), W₃(;3), W₃(;4), W₆(;1), W₆(;2), W₆(;3), and W₆(;4). In this example, W_(k)(;n) denotes the n^(th) column vector of W_(k).

The codewords included in the above Table 2 may be expressed equivalent as follows:

$M_{1} = \begin{matrix} 0.5000 \\ 0.5000 \\ 0.5000 \\ 0.5000 \end{matrix}$ $M_{2} = \begin{matrix} 0.5000 \\ {- 0.5000} \\ 0.5000 \\ {- 0.5000} \end{matrix}$ $M_{3} = \begin{matrix} {- 0.5000} \\ {0 - {0.5000{\mathbb{i}}}} \\ 0.5000 \\ {0 + {0.5000{\mathbb{i}}}} \end{matrix}$ $M_{4} = \begin{matrix} {- 0.5000} \\ {0 + {0.5000{\mathbb{i}}}} \\ 0.5000 \\ {0 - {0.5000{\mathbb{i}}}} \end{matrix}$

$M_{5} = \begin{matrix} 0.5000 \\ {0.3536 + {0.3536{\mathbb{i}}}} \\ {0 + {0.5000{\mathbb{i}}}} \\ {{- 0.3536} + {0.3536{\mathbb{i}}}} \end{matrix}$ $M_{6} = \begin{matrix} 0.5000 \\ {{- 0.3536} + {0.3536{\mathbb{i}}}} \\ {0 - {0.5000{\mathbb{i}}}} \\ {0.3536 + {0.3536{\mathbb{i}}}} \end{matrix}$ $M_{7} = \begin{matrix} 0.5000 \\ {{- 0.3536} - {0.3536{\mathbb{i}}}} \\ {0 + {0.5000{\mathbb{i}}}} \\ {0.3536 - {0.3536{\mathbb{i}}}} \end{matrix}$ $M_{8} = \begin{matrix} 0.5000 \\ {0.3536 - {0.3536{\mathbb{i}}}} \\ {0 - {0.5000{\mathbb{i}}}} \\ {{- 0.3536} - {0.3536{\mathbb{i}}}} \end{matrix}$

In this example, the codebook used in the multi-user MIMO communication system performing non-unitary precoding has a base station that includes four physical antennas. Codewords included in the codebook used in the multi-user MIMO communication system performing non-unitary precoding may be given by the following Table 3. In the multi-user MIMO communication system, a rank of each of users is 1:

Transmit Codebook Index Rank 1 1 C_(1,1)=W1(:,2) 2 C_(2,1)=W1(:,3) 3 C_(3,1)=W1(:,4) 4 C_(4,1)=W2(:,2) 5 C_(5,1)=W2(:,3) 6 C_(6,1)=W2(:,4) 7 C_(7,1)=W3(:,1) 8 C_(8,1)=W4(:,1) 9 C_(9,1)=W5(:,1) 10 C_(10,1)=W5(:,2) 11 C_(11,1)=W5(:,3) 12 C_(12,1)=W5(:,4) 13 C_(13,1)=W6(:,1) 14 C_(14,1)=W6(:,2) 15 C_(15,1)=W6(:,3) 16 C_(16,1)=W6(:,4)

The codewords included in the above Table 3 may be expressed equivalently as follows:

$C_{1,1} = \begin{matrix} 0.5000 \\ {- 0.5000} \\ 0.5000 \\ {- 0.5000} \end{matrix}$ $C_{2,1} = \begin{matrix} {- 0.5000} \\ {- 0.5000} \\ 0.5000 \\ 0.5000 \end{matrix}$ $C_{3,1} = \begin{matrix} {- 0.5000} \\ 0.5000 \\ 0.5000 \\ {- 0.5000} \end{matrix}$ $C_{4,1} = \begin{matrix} 0.5000 \\ {0 - {0.5000{\mathbb{i}}}} \\ 0.5000 \\ {0 - {0.5000{\mathbb{i}}}} \end{matrix}$

$C_{5,1} = \begin{matrix} {- 0.5000} \\ {0 - {0.5000{\mathbb{i}}}} \\ 0.5000 \\ {0 + {0.5000{\mathbb{i}}}} \end{matrix}$ $C_{6,1} = \begin{matrix} {- 0.5000} \\ {0 + {0.5000{\mathbb{i}}}} \\ 0.5000 \\ {0 - {0.5000{\mathbb{i}}}} \end{matrix}$ $C_{7,1} = \begin{matrix} 0.5000 \\ 0.5000 \\ 0.5000 \\ 0.5000 \end{matrix}$ $C_{8,1} = \begin{matrix} 0.5000 \\ {0 + {0.5000{\mathbb{i}}}} \\ 0.5000 \\ {0 + {0.5000{\mathbb{i}}}} \end{matrix}$

$C_{9,1} = \begin{matrix} 0.5000 \\ 0.5000 \\ 0.5000 \\ {- 0.5000} \end{matrix}$ $C_{10,1} = \begin{matrix} 0.5000 \\ {0 + {0.5000{\mathbb{i}}}} \\ {- 0.5000} \\ {0 + {0.5000{\mathbb{i}}}} \end{matrix}$ $C_{11,1} = \begin{matrix} 0.5000 \\ {- 0.5000} \\ 0.5000 \\ 0.5000 \end{matrix}$ $C_{12,1} = \begin{matrix} 0.5000 \\ {0 - {0.5000{\mathbb{i}}}} \\ {- 0.5000} \\ {0 - {0.5000{\mathbb{i}}}} \end{matrix}$

$C_{13,1} = \begin{matrix} 0.5000 \\ {03536 + {0.3536{\mathbb{i}}}} \\ {0 + {0.5000{\mathbb{i}}}} \\ {{- 0.3536} + {0.3536{\mathbb{i}}}} \end{matrix}$ $C_{14,1} = \begin{matrix} 0.5000 \\ {{- 03536} + {0.3536{\mathbb{i}}}} \\ {0 - {0.5000{\mathbb{i}}}} \\ {0.3536 + {0.3536{\mathbb{i}}}} \end{matrix}$ $C_{15,1} = \begin{matrix} 0.5000 \\ {{- 03536} - {0.3536{\mathbb{i}}}} \\ {0 + {0.5000{\mathbb{i}}}} \\ {0.3536 - {0.3536{\mathbb{i}}}} \end{matrix}$ $C_{16,1} = \begin{matrix} 0.5000 \\ {03536 - {0.3536{\mathbb{i}}}} \\ {0 - {0.5000{\mathbb{i}}}} \\ {{- 0.3536} - {0.3536{\mathbb{i}}}} \end{matrix}$

Illustrated below is a second example of a codebook used in a single user MIMO communication system or a multi-user MIMO communication system where a number of physical antennas of a base station is four.

In this example, a rotation matrix U_(rot) and a QPSK DFT matrix may be defined as follows:

$U_{rot} = {\frac{1}{\sqrt{2}}\begin{bmatrix} 1 & 0 & {- 1} & 0 \\ 0 & 1 & 0 & {- 1} \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \end{bmatrix}}$ and ${DFT} = {0.5*{\begin{bmatrix} 1 & 1 & 1 & 1 \\ 1 & j & {- 1} & {- j} \\ 1 & {- 1} & 1 & {- 1} \\ 1 & {- j} & {- 1} & j \end{bmatrix}.}}$ Also, diag(a, b, c, d) is a 4×4 matrix, and diagonal elements of diag(a, b, c, d) are a, b, c, and d, and all the remaining elements are zero.

Codewords W₁, W₂, W₃, W₄, W₅, and W₆ may be defined as follows:

$\begin{matrix} \begin{matrix} {W_{1} = {\frac{1}{\sqrt{2}}*U_{rot}*\begin{bmatrix} 1 & 1 & 0 & 0 \\ 1 & {- 1} & 0 & 0 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 1 & {- 1} \end{bmatrix}*\begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & {- 1} & 0 \\ 0 & 0 & 0 & {- 1} \end{bmatrix}}} \\ {= {0.5*\begin{bmatrix} 1 & 1 & 1 & 1 \\ 1 & {- 1} & 1 & {- 1} \\ 1 & 1 & {- 1} & {- 1} \\ 1 & {- 1} & {- 1} & 1 \end{bmatrix}}} \end{matrix} & \; \\ \begin{matrix} {W_{2} = {\frac{1}{\sqrt{2}}*U_{rot}*\begin{bmatrix} 1 & 1 & 0 & 0 \\ j & {- j} & 0 & 0 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & j & {- j} \end{bmatrix}*\begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & {- 1} & 0 \\ 0 & 0 & 0 & {- 1} \end{bmatrix}}} \\ {= {0.5*\begin{bmatrix} 1 & 1 & 1 & 1 \\ j & {- j} & j & {- j} \\ 1 & 1 & {- 1} & {- 1} \\ j & {- j} & {- j} & j \end{bmatrix}}} \end{matrix} & \; \\ \begin{matrix} {W_{3} = {\frac{1}{\sqrt{2}}*U_{rot}*\begin{bmatrix} 1 & 1 & 0 & 0 \\ 1 & {- 1} & 0 & 0 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & j & {- j} \end{bmatrix}*\begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & {- 1} & 0 \\ 0 & 0 & 0 & {- 1} \end{bmatrix}}} \\ {= {0.5*\begin{bmatrix} 1 & 1 & 1 & 1 \\ 1 & {- 1} & j & {- j} \\ 1 & 1 & {- 1} & {- 1} \\ 1 & {- 1} & {- j} & j \end{bmatrix}}} \end{matrix} & \; \\ \begin{matrix} {W_{4} = {\frac{1}{\sqrt{2}}*U_{rot}*\begin{bmatrix} 1 & 1 & 0 & 0 \\ j & {- j} & 0 & 0 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 1 & {- 1} \end{bmatrix}*\begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & {- 1} & 0 \\ 0 & 0 & 0 & {- 1} \end{bmatrix}}} \\ {= {0.5*\begin{bmatrix} 1 & 1 & 1 & 1 \\ j & {- j} & 1 & {- 1} \\ 1 & 1 & {- 1} & {- 1} \\ j & {- j} & {- 1} & 1 \end{bmatrix}}} \end{matrix} & \; \\ \begin{matrix} {W_{5} = {{{diag}\left( {1,1,1,{- 1}} \right)}*{DFT}}} \\ {= {0.5*\begin{bmatrix} 1 & 1 & 1 & 1 \\ 1 & j & {- 1} & {- j} \\ 1 & {- 1} & 1 & {- 1} \\ {- 1} & j & 1 & {- j} \end{bmatrix}}} \end{matrix} & \; \\ \begin{matrix} {W_{6} = {{{diag}\left( {1,\frac{\left( {1 + j} \right)}{\sqrt{2}},j,\frac{\left( {{- 1} + j} \right)}{\sqrt{2}}} \right)}*{DFT}}} \\ {= {0.5*\begin{bmatrix} 1 & 1 & 1 & 1 \\ \frac{\left( {1 + j} \right)}{\sqrt{2}} & \frac{\left( {{- 1} + j} \right)}{\sqrt{2}} & \frac{\left( {{- 1} - j} \right)}{\sqrt{2}} & \frac{\left( {1 - j} \right)}{\sqrt{2}} \\ j & {- j} & j & {- j} \\ \frac{\left( {{- 1} + j} \right)}{\sqrt{2}} & \frac{\left( {1 + j} \right)}{\sqrt{2}} & \frac{\left( {1 - j} \right)}{\sqrt{2}} & \frac{\left( {{- 1} - j} \right)}{\sqrt{2}} \end{bmatrix}}} \end{matrix} & \; \end{matrix}$

When the number of physical antennas of the base station is four, matrices or codewords included in the codebook used in the single user MIMO communication system according to the second example may be given by the following Table 4:

Transmit Codebook Index Rank 1 Rank 2 Rank 3 Rank 4 1 C_(1,1)=W1(:,2) C_(1,2)=W1(:,2 1) C_(1,3)=W1(:,1 2 3) C_(1,4)=W1(:,1 2 3 4) 2 C_(2,1)=W1(:,3) C_(2,2)=W1(:,3 1) C_(2,3)=W1(:,1 2 4) C_(2,4)=W2(:,1 2 3 4) 3 C_(3,1)=W1(:,4) C_(3,2)=W1(:,4 1) C_(3,3)=W1(:,1 3 4) C_(3,4)=W3(:,1 2 3 4) 4 C_(4,1)=W2(:,2) C_(4,2)=W1(:,2 3) C_(4,3)=W1(:,2 3 4) C_(4,4)=W4(:,1 2 3 4) 5 C_(5,1)=W2(:,3) C_(5,2)=W1(:,2 4) C_(5,3)=W2(:,1 2 3) C_(5,4)=W5(:,1 2 3 4) 6 C_(6,1)=W2(:,4) C_(6,2)=W1(:,3 4) C_(6,3)=W2(:,1 2 4) C_(6,4)=W6(:,1 2 3 4) 7 C_(7,1)=W3(:,1) C_(7,2)=W2(:,3 1) C_(7,3)=W2(:,1 3 4) n/a 8 C_(8,1)=W4(:,1) C_(8,2)=W2(:,4 1) C_(8,3)=W2(:,2 3 4) n/a 9 C_(9,1)=W5(:,1) C_(9,2)=W2(:,2 3) C_(9,3)=W5(:,1 2 3) n/a 10 C_(10,1)=W5(:,2) C_(10,2)=W2(:,2 4) C_(10,3)=W5(:,1 2 4) n/a 11 C_(11,1)=W5(:,3) C_(11,2)=W3(:,3 1) C_(11,3)=W5(:,1 3 4) n/a 12 C_(12,1)=W5(:,4) C_(12,2)=W3(:,4 1) C_(12,3)=W5(:,2 3 4) n/a 13 C_(13,1)=W6(:,1) C_(13,2)=W4(:,3 1) C_(13,3)=W6(:,1 2 3) n/a 14 C_(14,1)=W6(:,2) C_(14,2)=W4(:,4 1) C_(14,3)=W6(:,1 2 4) n/a 15 C_(15,1)=W6(:,3) C_(15,2)=W5(:,1 3) C_(15,3)=W6(:,1 3 4) n/a 16 C_(16,1)=W6(:,4) C_(16,2)=W6(:,2 4) C_(16,3)=W6(:,2 3 4) n/a

The codewords included in the above Table 4 may be expressed equivalently as follows:

$C_{1,1} = \begin{matrix} 0.5000 \\ {- 0.5000} \\ 0.5000 \\ {- 0.5000} \end{matrix}$ $C_{2,1} = \begin{matrix} 0.5000 \\ 0.5000 \\ {- 0.5000} \\ {- 0.5000} \end{matrix}$ $C_{3,1} = \begin{matrix} 0.5000 \\ {- 0.5000} \\ {- 0.5000} \\ 0.5000 \end{matrix}$ $C_{4,1} = \begin{matrix} 0.5000 \\ {0 - {0.5000{\mathbb{i}}}} \\ 0.5000 \\ {0 - {0.5000{\mathbb{i}}}} \end{matrix}$

$C_{5,1} = \begin{matrix} 0.5000 \\ {0 + {0.5000{\mathbb{i}}}} \\ {- 0.5000} \\ {0 - {0.5000{\mathbb{i}}}} \end{matrix}$ $C_{6,1} = \begin{matrix} 0.5000 \\ {0 - {0.5000{\mathbb{i}}}} \\ {- 0.5000} \\ {0 + {0.5000{\mathbb{i}}}} \end{matrix}$ $C_{7,1} = \begin{matrix} 0.5000 \\ 0.5000 \\ 0.5000 \\ 0.5000 \end{matrix}$ $C_{8,1} = \begin{matrix} 0.5000 \\ {0 + {0.5000{\mathbb{i}}}} \\ 0.5000 \\ {0 + {0.5000{\mathbb{i}}}} \end{matrix}$

$C_{9,1} = \begin{matrix} 0.5000 \\ 0.5000 \\ 0.5000 \\ {- 0.5000} \end{matrix}$ $C_{10,1} = \begin{matrix} 0.5000 \\ {0 + {0.5000{\mathbb{i}}}} \\ {- 0.5000} \\ {0 + {0.5000{\mathbb{i}}}} \end{matrix}$ $C_{11,1} = \begin{matrix} 0.5000 \\ {- 0.5000} \\ 0.5000 \\ 0.5000 \end{matrix}$ $C_{12,1} = \begin{matrix} 0.5000 \\ {0 - {0.5000{\mathbb{i}}}} \\ {- 0.5000} \\ {0 - {0.5000{\mathbb{i}}}} \end{matrix}$

$C_{13,1} = \begin{matrix} 0.5000 \\ {03536 + {0.3536{\mathbb{i}}}} \\ {0 + {0.5000{\mathbb{i}}}} \\ {{- 0.3536} + {0.3536{\mathbb{i}}}} \end{matrix}$ $C_{14,1} = \begin{matrix} 0.5000 \\ {{- 03536} + {0.3536{\mathbb{i}}}} \\ {0 - {0.5000{\mathbb{i}}}} \\ {0.3536 + {0.3536{\mathbb{i}}}} \end{matrix}$ $C_{15,1} = \begin{matrix} 0.5000 \\ {{- 03536} - {0.3536{\mathbb{i}}}} \\ {0 + {0.5000{\mathbb{i}}}} \\ {0.3536 - {0.3536{\mathbb{i}}}} \end{matrix}$ $C_{16,1} = \begin{matrix} 0.5000 \\ {03536 - {0.3536{\mathbb{i}}}} \\ {0 - {0.5000{\mathbb{i}}}} \\ {{- 0.3536} - {0.3536{\mathbb{i}}}} \end{matrix}$

$C_{1,2} = \begin{matrix} 0.5000 & 0.5000 \\ {- 0.5000} & 0.5000 \\ 0.5000 & 0.5000 \\ {- 0.5000} & 0.5000 \end{matrix}$ $C_{2,2} = \begin{matrix} 0.5000 & 0.5000 \\ 0.5000 & 0.5000 \\ {- 0.5000} & 0.5000 \\ {- 0.5000} & 0.5000 \end{matrix}$ $C_{3,2} = \begin{matrix} 0.5000 & 0.5000 \\ {- 0.5000} & 0.5000 \\ {- 0.5000} & 0.5000 \\ 0.5000 & 0.5000 \end{matrix}$ $C_{4,2} = \begin{matrix} 0.5000 & 0.5000 \\ {- 0.5000} & 0.5000 \\ 0.5000 & {- 0.5000} \\ {- 0.5000} & {- 0.5000} \end{matrix}$

$C_{5,2} = \begin{matrix} 0.5000 & 0.5000 \\ {- 0.5000} & {- 0.5000} \\ 0.5000 & {- 0.5000} \\ {- 0.5000} & 0.5000 \end{matrix}$ $C_{6,2} = \begin{matrix} 0.5000 & 0.5000 \\ 0.5000 & {- 0.5000} \\ {- 0.5000} & {- 0.5000} \\ {- 0.5000} & 0.5000 \end{matrix}$ $C_{7,2} = \begin{matrix} 0.5000 & 0.5000 \\ {0 + {0.5000{\mathbb{i}}}} & {{0 + {0.5000\;{\mathbb{i}}}}\ } \\ {- 0.5000} & 0.5000 \\ {0 - {0.5000{\mathbb{i}}}} & {0 + {0.5000\;{\mathbb{i}}}} \end{matrix}$ $C_{8,2} = \begin{matrix} 0.5000 & 0.5000 \\ {0 - {0.5000{\mathbb{i}}}} & {0 + {0.5000\;{\mathbb{i}}}} \\ {- 0.5000} & 0.5000 \\ {0 + {0.5000{\mathbb{i}}}} & {0 + {0.5000\;{\mathbb{i}}}} \end{matrix}$

$C_{9,2} = \begin{matrix} 0.5000 & 0.5000 \\ {0 - {0.5000{\mathbb{i}}}} & {0 + {0.5000\;{\mathbb{i}}}} \\ 0.5000 & {- 0.5000} \\ {0 - {0.5000{\mathbb{i}}}} & {0 - {0.5000\;{\mathbb{i}}}} \end{matrix}$ $C_{10,2} = \begin{matrix} 0.5000 & 0.5000 \\ {0 - {0.5000{\mathbb{i}}}} & {0 - {0.5000\;{\mathbb{i}}}} \\ 0.5000 & {- 0.5000} \\ {0 - {0.5000{\mathbb{i}}}} & {0 + {0.5000\;{\mathbb{i}}}} \end{matrix}$ $C_{11,2} = \begin{matrix} 0.5000 & 0.5000 \\ {0 + {0.5000{\mathbb{i}}}} & 0.5000 \\ {- 0.5000} & 0.5000 \\ {0 - {0.5000{\mathbb{i}}}} & 0.5000 \end{matrix}$ $C_{12,2} = \begin{matrix} 0.5000 & 0.5000 \\ {0 - {0.5000{\mathbb{i}}}} & 0.5000 \\ {- 0.5000} & 0.5000 \\ {0 + {0.5000{\mathbb{i}}}} & 0.5000 \end{matrix}$

$C_{13,2} = \begin{matrix} 0.5000 & 0.5000 \\ 0.5000 & {0 + {0.5000{\mathbb{i}}}} \\ {- 0.5000} & 0.5000 \\ {- 0.5000} & {0 + {0.5000{\mathbb{i}}}} \end{matrix}$ $C_{14,2} = \begin{matrix} 0.5000 & 0.5000 \\ {- 0.5000} & {0 + {0.5000{\mathbb{i}}}} \\ {- 0.5000} & 0.5000 \\ 0.5000 & {0 + {0.5000{\mathbb{i}}}} \end{matrix}$ $C_{15,2} = \begin{matrix} 0.5000 & 0.5000 \\ 0.5000 & {- 0.5000} \\ 0.5000 & 0.5000 \\ {- 0.5000} & 0.5000 \end{matrix}$ $C_{16,2} = \begin{matrix} 0.5000 & 0.5000 \\ {{- 0.3536} + {0.3536{\mathbb{i}}}} & {0.3536 - {0.3536{\mathbb{i}}}} \\ {0 - {0.5000{\mathbb{i}}}} & {0 - {0.5000{\mathbb{i}}}} \\ {0.3536 + {0.3536{\mathbb{i}}}} & {{- 0.3536} - {0.3536{\mathbb{i}}}} \end{matrix}$

$C_{1,3} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 \\ 0.5000 & {- 0.5000} & 0.5000 \\ 0.5000 & 0.5000 & {- 0.5000} \\ 0.5000 & {- 0.5000} & {- 0.5000} \end{matrix}$ $C_{2,3} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 \\ 0.5000 & {- 0.5000} & {- 0.5000} \\ 0.5000 & 0.5000 & {- 0.5000} \\ 0.5000 & {- 0.5000} & 0.5000 \end{matrix}$ $C_{3,3} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 \\ 0.5000 & 0.5000 & {- 0.5000} \\ 0.5000 & {- 0.5000} & {- 0.5000} \\ 0.5000 & {- 0.5000} & 0.5000 \end{matrix}$ $C_{4,3} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 \\ {- 0.5000} & 0.5000 & {- 0.5000} \\ 0.5000 & {- 0.5000} & {- 0.5000} \\ {- 0.5000} & {- 0.5000} & 0.5000 \end{matrix}$

$C_{5,3} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 \\ {{0 + {0.5000\;{\mathbb{i}}}}\;} & {0 - {0.5000{\mathbb{i}}}} & {0 + {0.5000\;{\mathbb{i}}}} \\ 0.5000 & 0.5000 & {- 0.5000} \\ {0 + {0.5000\;{\mathbb{i}}}} & {0 - {0.5000\;{\mathbb{i}}}} & {0 - {0.5000\;{\mathbb{i}}}} \end{matrix}$ $C_{6,3} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 \\ {0 + {0.5000\;{\mathbb{i}}}} & {0 - {0.5000\;{\mathbb{i}}}} & {0 - {0.5000\;{\mathbb{i}}}} \\ 0.5000 & 0.5000 & {- 0.5000} \\ {0 + {0.5000\;{\mathbb{i}}}} & {0 - {0.5000\;{\mathbb{i}}}} & {0 + {0.5000\;{\mathbb{i}}}} \end{matrix}$ $C_{7,3} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 \\ {0 + {0.5000\;{\mathbb{i}}}} & {0 + {0.5000\;{\mathbb{i}}}} & {0 - {0.5000\;{\mathbb{i}}}} \\ 0.5000 & {- 0.5000} & {- 0.5000} \\ {0 + {0.5000\;{\mathbb{i}}}} & {0 - {0.5000\;{\mathbb{i}}}} & {0 + {0.5000\;{\mathbb{i}}}} \end{matrix}$ $C_{8,3} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 \\ {0 - {0.5000\;{\mathbb{i}}}} & {0 + {0.5000\;{\mathbb{i}}}} & {0 - {0.5000\;{\mathbb{i}}}} \\ 0.5000 & {- 0.5000} & {- 0.5000} \\ {0 - {0.5000\;{\mathbb{i}}}} & {0 - {0.5000\;{\mathbb{i}}}} & {0 + {0.5000\;{\mathbb{i}}}} \end{matrix}$

$C_{9,3} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 \\ 0.5000 & {0 + {0.5000\;{\mathbb{i}}}} & {- 0.5000} \\ 0.5000 & {- 0.5000} & 0.5000 \\ {- 0.5000} & {0 + {0.5000\;{\mathbb{i}}}} & 0.5000 \end{matrix}$ $C_{10,3} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 \\ 0.5000 & {0 + {0.5000\;{\mathbb{i}}}} & {0 - {0.5000\;{\mathbb{i}}}} \\ 0.5000 & {- 0.5000} & {- 0.5000} \\ {- 0.5000} & {0 + {0.5000\;{\mathbb{i}}}} & {0 - {0.5000\;{\mathbb{i}}}} \end{matrix}$ $C_{11,3} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 \\ 0.5000 & {- 0.5000} & {0 - {0.5000\;{\mathbb{i}}}} \\ 0.5000 & 0.5000 & {- 0.5000} \\ {- 0.5000} & 0.5000 & {0 - {0.5000\;{\mathbb{i}}}} \end{matrix}$ $C_{12,3} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 \\ {0 + {0.5000{\mathbb{i}}}} & {- 0.5000} & {0 - {0.5000\;{\mathbb{i}}}} \\ {- 0.5000} & 0.5000 & {- 0.5000} \\ {0 + {0.5000{\mathbb{i}}}} & 0.5000 & {0 - {0.5000\;{\mathbb{i}}}} \end{matrix}$

$C_{13,3} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 \\ {0.3536 + {0.3536{\mathbb{i}}}} & {{- 0.3536} + {0.3536{\mathbb{i}}}} & {{- 0.3536} - {0.3536{\mathbb{i}}}} \\ {0 + {5000{\mathbb{i}}}} & {0 - {0.5000{\mathbb{i}}}} & {0 + {0.5000{\mathbb{i}}}} \\ {{- 0.3536} + {0.3536{\mathbb{i}}}} & {0.3536 + {0.3536{\mathbb{i}}}} & {0.3536 - {0.3536{\mathbb{i}}}} \end{matrix}$ $C_{14,3} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 \\ {0.3536 + {0.3536{\mathbb{i}}}} & {{- 0.3536} + {0.3536{\mathbb{i}}}} & {0.3536 - {0.3536{\mathbb{i}}}} \\ {0 + {5000{\mathbb{i}}}} & {0 - {0.5000{\mathbb{i}}}} & {0 - {0.5000{\mathbb{i}}}} \\ {{- 0.3536} + {0.3536{\mathbb{i}}}} & {0.3536 + {0.3536{\mathbb{i}}}} & {{- 0.3536} - {0.3536{\mathbb{i}}}} \end{matrix}$ $C_{15,3} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 \\ {0.3536 + {0.3536{\mathbb{i}}}} & {{- 0.3536} - {0.3536{\mathbb{i}}}} & {0.3536 - {0.3536{\mathbb{i}}}} \\ {0 + {5000{\mathbb{i}}}} & {0 + {0.5000{\mathbb{i}}}} & {0 - {0.5000{\mathbb{i}}}} \\ {{- 0.3536} + {0.3536{\mathbb{i}}}} & {0.3536 - {0.3536{\mathbb{i}}}} & {{- 0.3536} - {0.3536{\mathbb{i}}}} \end{matrix}$ $C_{16,3} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 \\ {{- 0.3536} + {0.3536{\mathbb{i}}}} & {{- 0.3536} - {0.3536{\mathbb{i}}}} & {0.3536 - {0.3536{\mathbb{i}}}} \\ {0 - {5000{\mathbb{i}}}} & {0 + {0.5000{\mathbb{i}}}} & {0 - {0.5000{\mathbb{i}}}} \\ {0.3536 + {0.3536{\mathbb{i}}}} & {0.3536 - {0.3536{\mathbb{i}}}} & {{- 0.3536} - {0.3536{\mathbb{i}}}} \end{matrix}$

$C_{1,4} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 & 0.5000 \\ 0.5000 & {- 0.5000} & 0.5000 & {- 0.5000} \\ 0.5000 & 0.5000 & {- 0.5000} & {- 0.5000} \\ 0.5000 & {- 0.5000} & {- 0.5000} & 0.5000 \end{matrix}$ $C_{2,4} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 & 0.5000 \\ {0 + {0.5000{\mathbb{i}}}} & {0 - {0.5000{\mathbb{i}}}} & {0 + {0.5000{\mathbb{i}}}} & {0 - {0.5000{\mathbb{i}}}} \\ 0.5000 & 0.5000 & {- 0.5000} & {- 0.5000} \\ {0 + {0.5000{\mathbb{i}}}} & {0 - {0.5000{\mathbb{i}}}} & {0 - {0.5000{\mathbb{i}}}} & {0 + {0.5000{\mathbb{i}}}} \end{matrix}$ $C_{3,4} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 & 0.5000 \\ 0.5000 & {- 0.5000} & {0 + {0.5000{\mathbb{i}}}} & {0 - {0.5000{\mathbb{i}}}} \\ 0.5000 & 0.5000 & {- 0.5000} & {- 0.5000} \\ 0.5000 & {- 0.5000} & {0 - {0.5000{\mathbb{i}}}} & {0 + {0.5000{\mathbb{i}}}} \end{matrix}$ $C_{4,4} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 & 0.5000 \\ {0 + {0.5000{\mathbb{i}}}} & {0 - {0.5000{\mathbb{i}}}} & 0.5000 & {- 0.5000} \\ 0.5000 & 0.5000 & {- 0.5000} & {- 0.5000} \\ {0 + {0.5000{\mathbb{i}}}} & {0 - {0.5000{\mathbb{i}}}} & {- 0.5000} & 0.5000 \end{matrix}$

$C_{5,4} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 & 0.5000 \\ 0.5000 & {0 + {0.5000{\mathbb{i}}}} & {- 0.5000} & {0 - {0.5000{\mathbb{i}}}} \\ 0.5000 & {- 0.5000} & 0.5000 & {- 0.5000} \\ {- 0.5000} & {0 + {0.5000{\mathbb{i}}}} & 0.5000 & {0 - {0.5000{\mathbb{i}}}} \end{matrix}$ $C_{6,4} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 & 0.5000 \\ {0.3536 + {0.3536{\mathbb{i}}}} & {{- 0.3536} + {0.3536{\mathbb{i}}}} & {{- 0.3536} - {0.3536{\mathbb{i}}}} & {0.3536 - {0.3536{\mathbb{i}}}} \\ {0 + {5000{\mathbb{i}}}} & {0 - {0.5000{\mathbb{i}}}} & {0 + {0.5000{\mathbb{i}}}} & {0 - {0.5000{\mathbb{i}}}} \\ {{- 0.3536} + {0.3536{\mathbb{i}}}} & {0.3536 + {0.3536{\mathbb{i}}}} & {0.3536 - {0.3536{\mathbb{i}}}} & {{- 0.3536} - {0.3536{\mathbb{i}}}} \end{matrix}$

For the second example, where the number of physical antennas of the base station is four, the codebook used in the multi-user MIMO communication system according to the second example, may be designed by combining subsets of two matrices, for example, W₃ and W₆ as follows:

$\begin{matrix} \begin{matrix} {W_{3} = {\frac{1}{\sqrt{2}}*U_{rot}*\begin{bmatrix} 1 & 1 & 0 & 0 \\ 1 & {- 1} & 0 & 0 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & j & {- j} \end{bmatrix}}} \\ {= {0.5*\begin{bmatrix} 1 & 1 & {- 1} & {- 1} \\ 1 & {- 1} & {- j} & j \\ 1 & 1 & 1 & 1 \\ 1 & {- 1} & j & {- j} \end{bmatrix}}} \end{matrix} & \; \\ \begin{matrix} {W_{6} = {{{diag}\left( {1,\frac{\left( {1 + j} \right)}{\sqrt{2}},j,\frac{\left( {{- 1} + j} \right)}{\sqrt{2}}} \right)}*{DFT}}} \\ {= {0.5*\begin{bmatrix} 1 & 1 & 1 & 1 \\ \frac{\left( {1 + j} \right)}{\sqrt{2}} & \frac{\left( {{- 1} + j} \right)}{\sqrt{2}} & \frac{\left( {{- 1} - j} \right)}{\sqrt{2}} & \frac{\left( {1 - j} \right)}{\sqrt{2}} \\ j & {- j} & j & {- j} \\ \frac{\left( {{- 1} + j} \right)}{\sqrt{2}} & \frac{\left( {1 + j} \right)}{\sqrt{2}} & \frac{\left( {1 - j} \right)}{\sqrt{2}} & \frac{\left( {{- 1} - j} \right)}{\sqrt{2}} \end{bmatrix}}} \end{matrix} & \; \end{matrix}$

When the number of physical antennas of the base station is four, the matrices or the codewords included in the codebook used in the multi-user MIMO communication system according to the second example may be given by the following Table 5:

Transmit Codebook Index Rank 1 1 M₁=W3(:,1) 2 M₂=W3(:,2) 3 M₃=W3(:,3) 4 M₄=W3(:,4) 5 M₅=W6(:,1) 6 M₆=W6(:,2) 7 M₇=W6(:,3) 8 M₈=W6(:,4)

The codewords included in the above Table 5 may be expressed equivalently as follows:

$M_{1} = \begin{matrix} 0.5000 \\ 0.5000 \\ 0.5000 \\ 0.5000 \end{matrix}$ $M_{2} = \begin{matrix} 0.5000 \\ {- 0.5000} \\ 0.5000 \\ {- 0.5000} \end{matrix}$ $M_{3} = \begin{matrix} {- 0.5000} \\ {0 - {0.5000{\mathbb{i}}}} \\ 0.5000 \\ {0 + {0.5000{\mathbb{i}}}} \end{matrix}$ $M_{4} = \begin{matrix} {- 0.5000} \\ {0 + {0.5000{\mathbb{i}}}} \\ 0.5000 \\ {0 - {0.5000{\mathbb{i}}}} \end{matrix}$

$M_{5} = \begin{matrix} 0.5000 \\ {0.3536 + {0.3536{\mathbb{i}}}} \\ {0 + {0.5000{\mathbb{i}}}} \\ {{- 0.3536} + {0.3536{\mathbb{i}}}} \end{matrix}$ $M_{6} = \begin{matrix} 0.5000 \\ {{- 0.3536} + {0.3536{\mathbb{i}}}} \\ {0 - {0.5000{\mathbb{i}}}} \\ {0.3536 + {0.3536{\mathbb{i}}}} \end{matrix}$ $M_{7} = \begin{matrix} 0.5000 \\ {{- 0.3536} - {0.3536{\mathbb{i}}}} \\ {0 + {0.5000{\mathbb{i}}}} \\ {0.3536 - {0.3536{\mathbb{i}}}} \end{matrix}$ $M_{8} = \begin{matrix} 0.5000 \\ {0.3536 - {0.3536{\mathbb{i}}}} \\ {0 - {0.5000{\mathbb{i}}}} \\ {{- 0.3536} - {0.3536{\mathbb{i}}}} \end{matrix}$

In this example, the codebook used in the multi-user MIMO communication system performing non-unitary precoding has a base station including four physical antennas.

Codewords included in the codebook used in the multi-user MIMO communication system performing non-unitary precoding may be given by the following Table 6:

Transmit Codebook Transmission Index Rank 1 1 C_(1,1)=W1(;,2) 2 C_(2,1)=W1(;,3) 3 C_(3,1)=W1(;,4) 4 C_(4,1)=W2(;,2) 5 C_(5,1)=W2(;,3) 6 C_(6,1)=W2(;,4) 7 C_(7,1)=W3(;,1) 8 C_(8,1)=W4(;,1) 9 C_(9,1)=W5(;,1) 10 C_(10,1)=W5(;,2) 11 C_(11,1)=W5(;,3) 12 C_(12,1)=W5(;,4) 13 C_(13,1)=W6(;,1) 14 C_(14,1)=W6(;,2) 15 C_(15,1)=W6(;,3) 16 C_(16,1)=W6(;,4)

The codewords included in the above Table 6 may be expressed equivalently as follows:

$C_{1,1} = \begin{matrix} 0.5000 \\ {- 0.5000} \\ 0.5000 \\ {- 0.5000} \end{matrix}$ $C_{2,1} = \begin{matrix} {- 0.5000} \\ {- 0.5000} \\ 0.5000 \\ 0.5000 \end{matrix}$ $C_{3,1} = \begin{matrix} {- 0.5000} \\ 0.5000 \\ 0.5000 \\ {- 0.5000} \end{matrix}$ $C_{4,1} = \begin{matrix} 0.5000 \\ {0 - {0.5000{\mathbb{i}}}} \\ 0.5000 \\ {0 - {0.5000{\mathbb{i}}}} \end{matrix}$

$C_{5,1} = \begin{matrix} {- 0.5000} \\ {0 - {0.5000{\mathbb{i}}}} \\ 0.5000 \\ {0 + {0.5000{\mathbb{i}}}} \end{matrix}$ $C_{6,1} = \begin{matrix} {- 0.5000} \\ {0 + {0.5000{\mathbb{i}}}} \\ 0.5000 \\ {0 - {0.5000{\mathbb{i}}}} \end{matrix}$ $C_{7,1} = \begin{matrix} 0.5000 \\ 0.5000 \\ 0.5000 \\ 0.5000 \end{matrix}$ $C_{8,1} = \begin{matrix} 0.5000 \\ {0 + {0.5000{\mathbb{i}}}} \\ 0.5000 \\ {0 + {0.5000{\mathbb{i}}}} \end{matrix}$

$C_{9,1} = \begin{matrix} 0.5000 \\ 0.5000 \\ 0.5000 \\ {- 0.5000} \end{matrix}$ $C_{10,1} = \begin{matrix} 0.5000 \\ {0 + {0.5000{\mathbb{i}}}} \\ {- 0.5000} \\ {0 + {0.5000{\mathbb{i}}}} \end{matrix}$ $C_{11,1} = \begin{matrix} 0.5000 \\ {- 0.5000} \\ 0.5000 \\ 0.5000 \end{matrix}$ $C_{12,1} = \begin{matrix} 0.5000 \\ {0 - {0.5000{\mathbb{i}}}} \\ {- 0.5000} \\ {0 - {0.5000{\mathbb{i}}}} \end{matrix}$

$C_{13,1} = \begin{matrix} 0.5000 \\ {03536 + {0.3536{\mathbb{i}}}} \\ {0 + {0.5000{\mathbb{i}}}} \\ {{- 0.3536} + {0.3536{\mathbb{i}}}} \end{matrix}$ $C_{14,1} = \begin{matrix} 0.5000 \\ {{- 03536} + {0.3536{\mathbb{i}}}} \\ {0 - {0.5000{\mathbb{i}}}} \\ {0.3536 + {0.3536{\mathbb{i}}}} \end{matrix}$ $C_{15,1} = \begin{matrix} 0.5000 \\ {{- 03536} - {0.3536{\mathbb{i}}}} \\ {0 + {0.5000{\mathbb{i}}}} \\ {0.3536 - {0.3536{\mathbb{i}}}} \end{matrix}$ $C_{16,1} = \begin{matrix} 0.5000 \\ {03536 - {0.3536{\mathbb{i}}}} \\ {0 - {0.5000{\mathbb{i}}}} \\ {{- 0.3536} - {0.3536{\mathbb{i}}}} \end{matrix}$

A third example of a codebook used in a single user MIMO communication system or a multi-user MIMO communication system where the number of physical antennas of a base station is four is illustrated below.

In this example, a rotation matrix U_(rot) and a QPSK DFT matrix may be defined as follows.

$U_{rot} = {\frac{1}{\sqrt{2}}\begin{bmatrix} 1 & 0 & {- 1} & 0 \\ 0 & 1 & 0 & {- 1} \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \end{bmatrix}}$ and ${DFT} = {0.5*{\begin{bmatrix} 1 & 1 & 1 & 1 \\ 1 & j & {- 1} & {- j} \\ 1 & {- 1} & 1 & {- 1} \\ 1 & {- j} & {- 1} & j \end{bmatrix}.}}$ Also, diag(a, b, c, d) is a 4×4 matrix, and diagonal elements of diag(a, b, c, d) are a, b, c, and d, and all the remaining elements are zero.

Codewords W₁, W₂, W₃, W₄, W₅, and W₆ may be defined as follows:

$\begin{matrix} \begin{matrix} {W_{1} = {\frac{1}{\sqrt{2}}*U_{rot}*\begin{bmatrix} 1 & 1 & 0 & 0 \\ 1 & {- 1} & 0 & 0 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 1 & {- 1} \end{bmatrix}}} \\ {= {0.5*\begin{bmatrix} 1 & 1 & {- 1} & {- 1} \\ 1 & {- 1} & {- 1} & 1 \\ 1 & 1 & 1 & 1 \\ 1 & {- 1} & 1 & {- 1} \end{bmatrix}}} \end{matrix} & \; \\ \begin{matrix} {W_{2} = {\frac{1}{\sqrt{2}}*U_{rot}*\begin{bmatrix} 1 & 1 & 0 & 0 \\ j & {- j} & 0 & 0 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & j & {- j} \end{bmatrix}}} \\ {= {0.5*\begin{bmatrix} 1 & 1 & {- 1} & {- 1} \\ j & {- j} & {- j} & j \\ 1 & 1 & 1 & 1 \\ j & {- j} & j & {- j} \end{bmatrix}}} \end{matrix} & \; \\ \begin{matrix} {W_{3} = {\frac{1}{\sqrt{2}}*U_{rot}*\begin{bmatrix} 1 & 1 & 0 & 0 \\ 1 & {- 1} & 0 & 0 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & j & {- j} \end{bmatrix}}} \\ {= {0.5*\begin{bmatrix} 1 & 1 & {- 1} & {- 1} \\ 1 & {- 1} & {- j} & j \\ 1 & 1 & 1 & 1 \\ 1 & {- 1} & j & {- j} \end{bmatrix}}} \end{matrix} & \; \\ \begin{matrix} {W_{4} = {\frac{1}{\sqrt{2}}*U_{rot}*\begin{bmatrix} 1 & 1 & 0 & 0 \\ j & {- j} & 0 & 0 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 1 & {- 1} \end{bmatrix}}} \\ {= {0.5*\begin{bmatrix} 1 & 1 & {- 1} & {- 1} \\ j & {- j} & {- 1} & 1 \\ 1 & 1 & 1 & 1 \\ j & {- j} & 1 & {- 1} \end{bmatrix}}} \end{matrix} & \; \\ \begin{matrix} {W_{5} = {{{diag}\left( {1,1,1,{- 1}} \right)}*{DFT}}} \\ {= {0.5*\begin{bmatrix} 1 & 1 & 1 & 1 \\ 1 & j & {- 1} & {- j} \\ 1 & {- 1} & 1 & {- 1} \\ {- 1} & j & 1 & {- j} \end{bmatrix}}} \end{matrix} & \; \\ \begin{matrix} {W_{6} = {{{diag}\left( {1,\frac{\left( {1 + j} \right)}{\sqrt{2}},j,\frac{\left( {{- 1} + j} \right)}{\sqrt{2}}} \right)}*{DFT}}} \\ {= {0.5*\begin{bmatrix} 1 & 1 & 1 & 1 \\ \frac{\left( {1 + j} \right)}{\sqrt{2}} & \frac{\left( {{- 1} + j} \right)}{\sqrt{2}} & \frac{\left( {{- 1} - j} \right)}{\sqrt{2}} & \frac{\left( {1 - j} \right)}{\sqrt{2}} \\ j & {- j} & j & {- j} \\ \frac{\left( {{- 1} + j} \right)}{\sqrt{2}} & \frac{\left( {1 + j} \right)}{\sqrt{2}} & \frac{\left( {1 - j} \right)}{\sqrt{2}} & \frac{\left( {{- 1} - j} \right)}{\sqrt{2}} \end{bmatrix}}} \end{matrix} & \; \end{matrix}$

For example, where the number of physical antennas of the base station is four, the matrices or the codewords included in the codebook used in the single user MIMO communication system according to the third example may be given by the following Table 7:

Transmit Codebook Index Rank 1 Rank 2 Rank 3 Rank 4 1 C_(1,1)=W1(:,2) C_(1,2)=W1(:,12) C_(1,3)=W1(:,1 2 3) C_(1,4)=W1(:,1 2 3 4) 2 C_(2,1)=W1(:,3) C_(2,2)=W1(:,13) C_(2,3)=W1(:,1 2 4) C_(2,4)=W2(:,1 2 3 4) 3 C_(3,1)=W1(:,4) C_(3,2)=W1(:,14) C_(3,3)=W1(:,1 3 4) C_(3,4)=W3(:,1 2 3 4) 4 C_(4,1)=W2(:,2) C_(4,2)=W1(:,23) C_(4,3)=W1(:,2 3 4) C_(4,4)=W4(:,1 2 3 4) 5 C_(5,1)=W2(:,3) C_(5,2)=W1(:,24) C_(5,3)=W2(:,1 2 3) C_(5,4)=W5(:,1 2 3 4) 6 C_(6,1)=W2(:,4) C_(6,2)=W1(:,34) C_(6,3)=W2(:,1 2 4) C_(6,4)=W6(:,1 2 3 4) 7 C_(7,1)=W3(:,1) C_(7,2)=W2(:,13) C_(7,3)=W2(:,1 3 4) n/a 8 C_(8,1)=W4(:,1) C_(8,2)=W2(:,14) C_(8,3)=W2(:,2 3 4) n/a 9 C_(9,1)=W5(:,1) C_(9,2)=W2(:,23) C_(9,3)=W5(:,1 2 3) n/a 10 C_(10,1)=W5(:,2) C_(10,2)=W2(:,24) C_(10,3)=W5(:,1 2 4) n/a 11 C_(11,1)=W5(:,3) C_(11,2)=W3(:,13) C_(11,3)=W5(:,1 3 4) n/a 12 C_(12,1)=W5(:,4) C_(12,2)=W3(:,14) C_(12,3)=W5(:,2 3 4) n/a 13 C_(13,1)=W6(:,1) C_(13,2)=W4(:,13) C_(13,3)=W6(:,1 2 3) n/a 14 C_(14,1)=W6(:,2) C_(14,2)=W4(:,14) C_(14,3)=W6(:,1 2 4) n/a 15 C_(15,1)=W6(:,3) C_(15,2)=W5(:,13) C_(15,3)=W6(:,1 3 4) n/a 16 C_(16,1)=W6(:,4) C_(16,2)=W6(:,24) C_(16,3)=W6(:,2 3 4) n/a

The codewords included in the above Table 7 may be expressed equivalently as follows:

$C_{1,1} = \begin{matrix} 0.5000 \\ {- 0.5000} \\ 0.5000 \\ {- 0.5000} \end{matrix}$ $C_{2,1} = \begin{matrix} {- 0.5000} \\ {- 0.5000} \\ 0.5000 \\ 0.5000 \end{matrix}$ $C_{3,1} = \begin{matrix} {- 0.5000} \\ 0.5000 \\ 0.5000 \\ {- 0.5000} \end{matrix}$ $C_{4,1} = \begin{matrix} 0.5000 \\ {0 - {0.5000{\mathbb{i}}}} \\ 0.5000 \\ {0 - {0.5000{\mathbb{i}}}} \end{matrix}$

$C_{5,1} = \begin{matrix} {- 0.5000} \\ {0 - {0.5000{\mathbb{i}}}} \\ 0.5000 \\ {0 + {0.5000{\mathbb{i}}}} \end{matrix}$ $C_{6,1} = \begin{matrix} {- 0.5000} \\ {0 + {0.5000{\mathbb{i}}}} \\ 0.5000 \\ {0 - {0.5000{\mathbb{i}}}} \end{matrix}$ $C_{7,1} = \begin{matrix} 0.5000 \\ 0.5000 \\ 0.5000 \\ 0.5000 \end{matrix}$ $C_{8,1} = \begin{matrix} 0.5000 \\ {0 + {0.5000{\mathbb{i}}}} \\ 0.5000 \\ {0 + {0.5000{\mathbb{i}}}} \end{matrix}$

$C_{9,1} = \begin{matrix} 0.5000 \\ 0.5000 \\ 0.5000 \\ {- 0.5000} \end{matrix}$ $C_{10,1} = \begin{matrix} 0.5000 \\ {0 + {0.5000{\mathbb{i}}}} \\ {- 0.5000} \\ {0 + {0.5000{\mathbb{i}}}} \end{matrix}$ $C_{11,1} = \begin{matrix} 0.5000 \\ {- 0.5000} \\ 0.5000 \\ 0.5000 \end{matrix}$ $C_{12,1} = \begin{matrix} 0.5000 \\ {0 - {0.5000{\mathbb{i}}}} \\ {- 0.5000} \\ {0 - {0.5000{\mathbb{i}}}} \end{matrix}$

$C_{13,1} = \begin{matrix} 0.5000 \\ {0.3536 + {0.3536{\mathbb{i}}}} \\ {0 - {0.5000{\mathbb{i}}}} \\ {{- 0.3536} + {0.3536{\mathbb{i}}}} \end{matrix}$ $C_{14,1} = \begin{matrix} 0.5000 \\ {{- 0.3536} + {0.3536{\mathbb{i}}}} \\ {0 - {0.5000{\mathbb{i}}}} \\ {0.3536 + {0.3536{\mathbb{i}}}} \end{matrix}$ $C_{15,1} = \begin{matrix} 0.5000 \\ {{- 0.3536} - {0.3536{\mathbb{i}}}} \\ {0 + {0.5000{\mathbb{i}}}} \\ {0.3536 - {0.3536{\mathbb{i}}}} \end{matrix}$ $C_{16,1} = \begin{matrix} 0.5000 \\ {0.3536 - {0.3536{\mathbb{i}}}} \\ {0 - {0.5000{\mathbb{i}}}} \\ {{- 0.3536} - {0.3536{\mathbb{i}}}} \end{matrix}$

$C_{1,2} = \begin{matrix} 0.5000 & 0.5000 \\ 0.5000 & {- 0.5000} \\ 0.5000 & 0.5000 \\ 0.5000 & {- 0.5000} \end{matrix}$ $C_{2,2} = \begin{matrix} 0.5000 & {- 0.5000} \\ 0.5000 & {- 0.5000} \\ 0.5000 & 0.5000 \\ 0.5000 & 0.5000 \end{matrix}$ $C_{3,2} = \begin{matrix} 0.5000 & {- 0.5000} \\ 0.5000 & 0.5000 \\ 0.5000 & 0.5000 \\ 0.5000 & {- 0.5000} \end{matrix}$ $C_{4,2} = \begin{matrix} 0.5000 & {- 0.5000} \\ {- 0.5000} & {- 0.5000} \\ 0.5000 & 0.5000 \\ {- 0.5000} & 0.5000 \end{matrix}$

$C_{5.2} = \begin{matrix} 0.5000 & {- 0.5000} \\ {- 0.5000} & 0.5000 \\ 0.5000 & 0.5000 \\ {- 0.5000} & {- 0.5000} \end{matrix}$ $C_{6,2} = \begin{matrix} {- 0.5000} & {- 0.5000} \\ {- 0.5000} & 0.5000 \\ 0.5000 & 0.5000 \\ 0.5000 & {- 0.5000} \end{matrix}$ $C_{7,2} = \begin{matrix} 0.5000 & {- 0.5000} \\ {0 + {0.5000\;{\mathbb{i}}}} & {0 - {0.5000\;{\mathbb{i}}}} \\ 0.5000 & 0.5000 \\ {0 + {0.5000\;{\mathbb{i}}}} & {0 + {0.5000\;{\mathbb{i}}}} \end{matrix}$ $C_{8,2} = \begin{matrix} 0.5000 & {- 0.5000} \\ {0 + {0.5000\;{\mathbb{i}}}} & {0 + {0.5000\;{\mathbb{i}}}} \\ 0.5000 & 0.5000 \\ {0 + {0.5000\;{\mathbb{i}}}} & {0 - {0.5000\;{\mathbb{i}}}} \end{matrix}$ $C_{9,2} = \begin{matrix} 0.5000 & {- 0.5000} \\ {0 - {0.5000\;{\mathbb{i}}}} & {0 - {0.5000\;{\mathbb{i}}}} \\ 0.5000 & 0.5000 \\ {0 - {0.5000\;{\mathbb{i}}}} & {0 + {0.5000\;{\mathbb{i}}}} \end{matrix}$

$C_{10,2} = \begin{matrix} 0.5000 & {- 0.5000} \\ {0 - {0.5000\;{\mathbb{i}}}} & {0 + {0.5000\;{\mathbb{i}}}} \\ 0.5000 & 0.5000 \\ {0 - {0.5000\;{\mathbb{i}}}} & {0 - {0.5000\;{\mathbb{i}}}} \end{matrix}$ $C_{11,2} = \begin{matrix} 0.5000 & {- 0.5000} \\ 0.5000 & {0 - {0.5000\;{\mathbb{i}}}} \\ 0.5000 & 0.5000 \\ 0.5000 & {0 + {0.5000\;{\mathbb{i}}}} \end{matrix}$ $C_{12,2} = \begin{matrix} 0.5000 & {- 0.5000} \\ 0.5000 & {0 + {0.5000\;{\mathbb{i}}}} \\ 0.5000 & 0.5000 \\ 0.5000 & {0 - {0.5000\;{\mathbb{i}}}} \end{matrix}$ $C_{13,2} = \begin{matrix} 0.5000 & {- 0.5000} \\ {0 + {0.5000\;{\mathbb{i}}}} & {- 0.5000} \\ 0.5000 & 0.5000 \\ {0 + {0.5000\;{\mathbb{i}}}} & 0.5000 \end{matrix}$ $C_{14,2} = \begin{matrix} 0.5000 & {- 0.5000} \\ {0 + {0.5000\;{\mathbb{i}}}} & 0.5000 \\ 0.5000 & 0.5000 \\ {0 + {0.5000\;{\mathbb{i}}}} & {- 0.5000} \end{matrix}$

$C_{15,2} = \begin{matrix} 0.5000 & 0.5000 \\ 0.5000 & {- 0.5000} \\ 0.5000 & 0.5000 \\ {- 0.5000} & 0.5000 \end{matrix}$ $C_{16,2} = \begin{matrix} 0.5000 & 0.5000 \\ {{- 0.3536} + {0.3536\;{\mathbb{i}}}} & {0.3536 - {0.3536\;{\mathbb{i}}}} \\ {0 - {0.5000\;{\mathbb{i}}}} & {0 - {0.5000\;{\mathbb{i}}}} \\ {0.3536 + {0.3536\;{\mathbb{i}}}} & {{- 0.3536} - {0.3536\;{\mathbb{i}}}} \end{matrix}$ $C_{1,3} = \begin{matrix} 0.5000 & 0.5000 & {- 0.5000} \\ 0.5000 & {- 0.5000} & {- 0.5000} \\ 0.5000 & 0.5000 & 0.5000 \\ 0.5000 & {- 0.5000} & 0.5000 \end{matrix}$ $C_{2,3} = \begin{matrix} 0.5000 & 0.5000 & {- 0.5000} \\ 0.5000 & {- 0.5000} & 0.5000 \\ 0.5000 & 0.5000 & 0.5000 \\ 0.5000 & {- 0.5000} & {- 0.5000} \end{matrix}$

$C_{3,3} = \begin{matrix} 0.5000 & {- 0.5000} & {- 0.5000} \\ 0.5000 & {- 0.5000} & 0.5000 \\ 0.5000 & 0.5000 & 0.5000 \\ 0.5000 & 0.5000 & {- 0.5000} \end{matrix}$ $C_{4,3} = \begin{matrix} 0.5000 & {- 0.5000} & {- 0.5000} \\ {- 0.5000} & {- 0.5000} & 0.5000 \\ 0.5000 & 0.5000 & 0.5000 \\ {- 0.5000} & 0.5000 & {- 0.5000} \end{matrix}$ $C_{5,3} = \begin{matrix} 0.5000 & 0.5000 & {- 0.5000} \\ {{0 + {0.5000\;{\mathbb{i}}}}\;} & {0 - {0.5000{\mathbb{i}}}} & {0 - {0.5000\;{\mathbb{i}}}} \\ 0.5000 & 0.5000 & 0.5000 \\ {0 + {0.5000\;{\mathbb{i}}}} & {0 - {0.5000\;{\mathbb{i}}}} & {0 + {0.5000\;{\mathbb{i}}}} \end{matrix}$ $C_{6,3} = \begin{matrix} 0.5000 & 0.5000 & {- 0.5000} \\ {0 + {0.5000\;{\mathbb{i}}}} & {0 - {0.5000\;{\mathbb{i}}}} & {0 + {0.5000\;{\mathbb{i}}}} \\ 0.5000 & 0.5000 & 0.5000 \\ {0 + {0.5000\;{\mathbb{i}}}} & {0 - {0.5000\;{\mathbb{i}}}} & {0 - {0.5000\;{\mathbb{i}}}} \end{matrix}$

$C_{7,3} = \begin{matrix} 0.5000 & {- 0.5000} & {- 0.5000} \\ {0 + {0.5000\;{\mathbb{i}}}} & {0 - {0.5000\;{\mathbb{i}}}} & {0 + {0.5000\;{\mathbb{i}}}} \\ 0.5000 & 0.5000 & 0.5000 \\ {0 + {0.5000\;{\mathbb{i}}}} & {0 + {0.5000\;{\mathbb{i}}}} & {0 - {0.5000\;{\mathbb{i}}}} \end{matrix}$ $C_{8,3} = \begin{matrix} 0.5000 & {- 0.5000} & {- 0.5000} \\ {0 - {0.5000\;{\mathbb{i}}}} & {0 - {0.5000\;{\mathbb{i}}}} & {0 + {0.5000\;{\mathbb{i}}}} \\ 0.5000 & 0.5000 & 0.5000 \\ {0 - {0.5000\;{\mathbb{i}}}} & {0 + {0.5000\;{\mathbb{i}}}} & {0 - {0.5000\;{\mathbb{i}}}} \end{matrix}$ $C_{9,3} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 \\ 0.5000 & {0 + {0.5000\;{\mathbb{i}}}} & {- 0.5000} \\ 0.5000 & {- 0.5000} & 0.5000 \\ {- 0.5000} & {0 + {0.5000\;{\mathbb{i}}}} & 0.5000 \end{matrix}$ $C_{10,3} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 \\ 0.5000 & {0 + {0.5000\;{\mathbb{i}}}} & {0 - {0.5000\;{\mathbb{i}}}} \\ 0.5000 & {- 0.5000} & {- 0.5000} \\ {- 0.5000} & {0 + {0.5000\;{\mathbb{i}}}} & {0 - {0.5000\;{\mathbb{i}}}} \end{matrix}$

$\mspace{20mu}{C_{11,3} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 \\ 0.5000 & {- 0.5000} & {0 - {0.5000\;{\mathbb{i}}}} \\ 0.5000 & 0.5000 & {- 0.5000} \\ {- 0.5000} & 0.5000 & {0 - {0.5000\;{\mathbb{i}}}} \end{matrix}}$ $\mspace{20mu}{C_{12,3} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 \\ {0 + {0.5000{\mathbb{i}}}} & {- 0.5000} & {0 - {0.5000\;{\mathbb{i}}}} \\ {- 0.5000} & 0.5000 & {- 0.5000} \\ {0 + {0.5000{\mathbb{i}}}} & 0.5000 & {0 - {0.5000\;{\mathbb{i}}}} \end{matrix}}$ $C_{13,3} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 \\ {0.3536 + {0.3536{\mathbb{i}}}} & {{- 0.3536} + {0.3536{\mathbb{i}}}} & {{- 0.3536} - {0.3536{\mathbb{i}}}} \\ {0 + {0.5000{\mathbb{i}}}} & {0 - {0.5000{\mathbb{i}}}} & {0 + {0.5000{\mathbb{i}}}} \\ {{- 0.3536} + {0.3536{\mathbb{i}}}} & {0.3536 + {0.3536{\mathbb{i}}}} & {0.3536 - {0.3536{\mathbb{i}}}} \end{matrix}$ $C_{14,3} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 \\ {0.3536 + {0.3536{\mathbb{i}}}} & {{- 0.3536} + {0.3536{\mathbb{i}}}} & {0.3536 - {0.3536{\mathbb{i}}}} \\ {0 + {0.5000{\mathbb{i}}}} & {0 - {0.5000{\mathbb{i}}}} & {0 - {0.5000{\mathbb{i}}}} \\ {{- 0.3536} + {0.3536{\mathbb{i}}}} & {0.3536 + {0.3536{\mathbb{i}}}} & {{- 0.3536} - {0.3536{\mathbb{i}}}} \end{matrix}$

$C_{15,3} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 \\ {0.3536 + {0.3536{\mathbb{i}}}} & {{- 0.3536} - {0.3536{\mathbb{i}}}} & {0.3536 - {0.3536{\mathbb{i}}}} \\ {0 + {0.5000{\mathbb{i}}}} & {0 + {0.5000{\mathbb{i}}}} & {0 - {0.5000{\mathbb{i}}}} \\ {{- 0.3536} + {0.3536{\mathbb{i}}}} & {0.3536 - {0.3536{\mathbb{i}}}} & {{- 0.3536} - {0.3536{\mathbb{i}}}} \end{matrix}$ $C_{16,3} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 \\ {{- 0.3536} + {0.3536{\mathbb{i}}}} & {{- 0.3536} - {0.3536{\mathbb{i}}}} & {0.3536 - {0.3536{\mathbb{i}}}} \\ {0 - {0.5000{\mathbb{i}}}} & {0 + {0.5000{\mathbb{i}}}} & {0 - {0.5000{\mathbb{i}}}} \\ {0.3536 + {0.3536{\mathbb{i}}}} & {0.3536 - {0.3536{\mathbb{i}}}} & {{- 0.3536} - {0.3536{\mathbb{i}}}} \end{matrix}$ $\mspace{20mu}{C_{1,4} = \begin{matrix} 0.5000 & 0.5000 & {- 0.5000} & {- 0.5000} \\ 0.5000 & {- 0.5000} & {- 0.5000} & 0.5000 \\ 0.5000 & 0.5000 & 0.5000 & 0.5000 \\ 0.5000 & {- 0.5000} & 0.5000 & {- 0.5000} \end{matrix}}$ $\mspace{20mu}{C_{2,4} = \begin{matrix} 0.5000 & 0.5000 & {- 0.5000} & {- 0.5000} \\ {0 + {0.5000{\mathbb{i}}}} & {0 - {0.5000{\mathbb{i}}}} & {0 - {0.5000{\mathbb{i}}}} & {0 + {0.5000{\mathbb{i}}}} \\ 0.5000 & 0.5000 & 0.5000 & 0.5000 \\ {0 + {0.5000{\mathbb{i}}}} & {0 - {0.5000{\mathbb{i}}}} & {0 + {0.5000{\mathbb{i}}}} & {0 - {0.5000{\mathbb{i}}}} \end{matrix}}$

$C_{3,4} = \begin{matrix} 0.5000 & 0.5000 & {- 0.5000} & {- 0.5000} \\ 0.5000 & {- 0.5000} & {0 - {0.5000{\mathbb{i}}}} & {0 + {0.5000{\mathbb{i}}}} \\ 0.5000 & 0.5000 & 0.5000 & 0.5000 \\ 0.5000 & {- 0.5000} & {0 + {0.5000{\mathbb{i}}}} & {0 - {0.5000{\mathbb{i}}}} \end{matrix}$ $C_{4,4} = \begin{matrix} 0.5000 & 0.5000 & {- 0.5000} & {- 0.5000} \\ {0 + {0.5000{\mathbb{i}}}} & {0 - {0.5000{\mathbb{i}}}} & {- 0.5000} & 0.5000 \\ 0.5000 & 0.5000 & 0.5000 & 0.5000 \\ {0 + {0.5000{\mathbb{i}}}} & {0 - {0.5000{\mathbb{i}}}} & 0.5000 & {- 0.5000} \end{matrix}$ $C_{5,4} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 & 0.5000 \\ 0.5000 & {0 + {0.5000{\mathbb{i}}}} & {- 0.5000} & {0 - {0.5000{\mathbb{i}}}} \\ 0.5000 & {- 0.5000} & 0.5000 & {- 0.5000} \\ {- 0.5000} & {0 + {0.5000{\mathbb{i}}}} & 0.5000 & {0 - {0.5000{\mathbb{i}}}} \end{matrix}$ $C_{6,4} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 & 0.5000 \\ {0.3536 + {0.3536{\mathbb{i}}}} & {{- 0.3536} + {0.3536{\mathbb{i}}}} & {{- 0.3536} - {0.3536{\mathbb{i}}}} & {0.3536 - {0.3536{\mathbb{i}}}} \\ {0 + {0.5000{\mathbb{i}}}} & {0 - {0.5000{\mathbb{i}}}} & {0 + {0.5000{\mathbb{i}}}} & {0 - {0.5000{\mathbb{i}}}} \\ {{- 0.3536} + {0.3536{\mathbb{i}}}} & {0.3536 + {0.3536{\mathbb{i}}}} & {0.3536 - {0.3536{\mathbb{i}}}} & {{- 0.3536} - {0.3536{\mathbb{i}}}} \end{matrix}$

For the third example, where the number of physical antennas of the base station is four, the codebook used in the multi-user MIMO communication system according to the third example may be designed by combining subsets of two matrices W₃ and W₆ as follows:

$\begin{matrix} \begin{matrix} {W_{3} = {\frac{1}{\sqrt{2}}*U_{rot}*\begin{bmatrix} 1 & 1 & 0 & 0 \\ 1 & {- 1} & 0 & 0 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & j & {- j} \end{bmatrix}}} \\ {= {0.5*\begin{bmatrix} 1 & 1 & {- 1} & {- 1} \\ 1 & {- 1} & {- j} & j \\ 1 & 1 & 1 & 1 \\ 1 & {- 1} & j & {- j} \end{bmatrix}}} \end{matrix} & \; \\ \begin{matrix} {W_{6} = {{{diag}\left( {1,\frac{\left( {1 + j} \right)}{\sqrt{2}},j,\frac{\left( {{- 1} + j} \right)}{\sqrt{2}}} \right)}*{DFT}}} \\ {= {0.5*\begin{bmatrix} 1 & 1 & 1 & 1 \\ \frac{\left( {1 + j} \right)}{\sqrt{2}} & \frac{\left( {{- 1} + j} \right)}{\sqrt{2}} & \frac{\left( {{- 1} - j} \right)}{\sqrt{2}} & \frac{\left( {1 - j} \right)}{\sqrt{2}} \\ j & {- j} & j & {- j} \\ \frac{\left( {{- 1} + j} \right)}{\sqrt{2}} & \frac{\left( {1 + j} \right)}{\sqrt{2}} & \frac{\left( {1 - j} \right)}{\sqrt{2}} & \frac{\left( {{- 1} - j} \right)}{\sqrt{2}} \end{bmatrix}}} \end{matrix} & \; \end{matrix}$

For example, where the number of physical antennas of the base station is four, the matrices or the codewords included in the codebook used in the multi-user MIMO communication system according to the third example may be expressed using a transmission rank, as given by the following Table 8:

Transmit Codebook Index Rank 1 1 M₁=W3(:,1) 2 M₂=W3(:,2) 3 M₃=W3(:,3) 4 M₄=W3(:,4) 5 M₅=W6(:,1) 6 M₆=W6(:,2) 7 M₇=W6(:,3) 8 M₈=W6(:,4)

The codewords included in the above Table 8 may be expressed equivalently as follows:

$M_{1} = \begin{matrix} 0.5000 \\ 0.5000 \\ 0.5000 \\ 0.5000 \end{matrix}$ $M_{2} = \begin{matrix} 0.5000 \\ {- 0.5000} \\ 0.5000 \\ {- 0.5000} \end{matrix}$ $M_{3} = \begin{matrix} {- 0.5000} \\ {0 - {0.5000{\mathbb{i}}}} \\ 0.5000 \\ {0 + {0.5000{\mathbb{i}}}} \end{matrix}$ $M_{4} = \begin{matrix} {- 0.5000} \\ {0 + {0.5000{\mathbb{i}}}} \\ 0.5000 \\ {0 - {0.5000{\mathbb{i}}}} \end{matrix}$

$M_{5} = \begin{matrix} 0.5000 \\ {0.3536 + {0.3536{\mathbb{i}}}} \\ {0 + {0.5000{\mathbb{i}}}} \\ {{- 0.3536} + {0.3536{\mathbb{i}}}} \end{matrix}$ $M_{6} = \begin{matrix} 0.5000 \\ {{- 0.3536} + {0.3536{\mathbb{i}}}} \\ {0 - {0.5000{\mathbb{i}}}} \\ {0.3536 + {0.3536{\mathbb{i}}}} \end{matrix}$ $M_{7} = \begin{matrix} 0.5000 \\ {{- 0.3536} - {0.3536{\mathbb{i}}}} \\ {0 + {0.5000{\mathbb{i}}}} \\ {0.3536 - {0.3536{\mathbb{i}}}} \end{matrix}$ $M_{8} = \begin{matrix} 0.5000 \\ {0.3536 - {0.3536{\mathbb{i}}}} \\ {0 - {0.5000{\mathbb{i}}}} \\ {{- 0.3536} - {0.3536{\mathbb{i}}}} \end{matrix}$

In this example, the codebook used in the multi-user MIMO communication system performing non-unitary precoding has a base station that includes four physical antennas.

Codewords included in the codebook used in the multi-user MIMO communication system performing non-unitary precoding may be given by the following Table 9:

Transmit Codebook Transmission Index Rank 1 1 C_(1,1)=W1(;,2) 2 C_(2,1)=W1(;,3) 3 C_(3,1)=W1(;,4) 4 C_(4,1)=W2(;,2) 5 C_(5,1)=W2(;,3) 6 C_(6,1)=W2(;,4) 7 C_(7,1)=W3(;,1) 8 C_(8,1)=W4(;,1) 9 C_(9,1)=W5(;,1) 10 C_(10,1)=W5(;,2) 11 C_(11,1)=W5(;,3) 12 C_(12,1)=W5(;,4) 13 C_(13,1)=W6(;,1) 14 C_(14,1)=W6(;,2) 15 C_(15,1)=W6(;,3) 16 C_(16,1)=W6(;,4)

The codewords included in the above Table 9 may be expressed equivalently as follows:

$C_{1,1} = \begin{matrix} 0.5000 \\ {- 0.5000} \\ 0.5000 \\ {- 0.5000} \end{matrix}$ $C_{2,1} = \begin{matrix} {- 0.5000} \\ {- 0.5000} \\ 0.5000 \\ 0.5000 \end{matrix}$ $C_{3,1} = \begin{matrix} {- 0.5000} \\ 0.5000 \\ 0.5000 \\ {- 0.5000} \end{matrix}$ $C_{4,1} = \begin{matrix} 0.5000 \\ {0 - {0.5000{\mathbb{i}}}} \\ 0.5000 \\ {0 - {0.5000{\mathbb{i}}}} \end{matrix}$

$C_{5,1} = \begin{matrix} {- 0.5000} \\ {0 - {0.5000{\mathbb{i}}}} \\ 0.5000 \\ {0 + {0.5000{\mathbb{i}}}} \end{matrix}$ $C_{6,1} = \begin{matrix} {- 0.5000} \\ {0 + {0.5000{\mathbb{i}}}} \\ 0.5000 \\ {0 - {0.5000{\mathbb{i}}}} \end{matrix}$ $C_{7,1} = \begin{matrix} 0.5000 \\ 0.5000 \\ 0.5000 \\ 0.5000 \end{matrix}$ $C_{8,1} = \begin{matrix} 0.5000 \\ {0 + {0.5000{\mathbb{i}}}} \\ 0.5000 \\ {0 + {0.5000{\mathbb{i}}}} \end{matrix}$

$C_{9,1} = \begin{matrix} 0.5000 \\ 0.5000 \\ 0.5000 \\ {- 0.5000} \end{matrix}$ $C_{10,1} = \begin{matrix} 0.5000 \\ {0 + {0.5000{\mathbb{i}}}} \\ {- 0.5000} \\ {0 + {0.5000{\mathbb{i}}}} \end{matrix}$ $C_{11,1} = \begin{matrix} 0.5000 \\ {- 0.5000} \\ 0.5000 \\ 0.5000 \end{matrix}$ $C_{12,1} = \begin{matrix} 0.5000 \\ {0 - {0.5000{\mathbb{i}}}} \\ {- 0.5000} \\ {0 - {0.5000{\mathbb{i}}}} \end{matrix}$

$C_{13,1} = \begin{matrix} 0.5000 \\ {0.3536 + {0.3536{\mathbb{i}}}} \\ {0 + {0.5000{\mathbb{i}}}} \\ {{- 0.3536} + {0.3536{\mathbb{i}}}} \end{matrix}$ $C_{14,1} = \begin{matrix} 0.5000 \\ {{- 0.3536} + {0.3536{\mathbb{i}}}} \\ {0 - {0.5000{\mathbb{i}}}} \\ {0.3536 + {0.3536{\mathbb{i}}}} \end{matrix}$ $C_{15,1} = \begin{matrix} 0.5000 \\ {{- 0.3536} - {03536{\mathbb{i}}}} \\ {0 + {0.5000{\mathbb{i}}}} \\ {0.3536 - {0.3536{\mathbb{i}}}} \end{matrix}$ $C_{16,1} = \begin{matrix} 0.5000 \\ {0.3536 - {0.3536{\mathbb{i}}}} \\ {0 - {0.5000{\mathbb{i}}}} \\ {{- 0.3536} - {0.3536{\mathbb{i}}}} \end{matrix}$

A fourth example of a codebook used in a single user MIMO communication system or a multi-user MIMO communication system where a number of physical antennas of a base station is four, is illustrated below.

In this example, a rotation matrix U_(rot) and a QPSK DFT matrix may be defined as follows.

$U_{rot} = {\frac{1}{\sqrt{2}}\begin{bmatrix} 1 & 0 & {- 1} & 0 \\ 0 & 1 & 0 & {- 1} \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \end{bmatrix}}$ and ${DFT} = {0.5*{\begin{bmatrix} 1 & 1 & 1 & 1 \\ 1 & j & {- 1} & {- j} \\ 1 & {- 1} & 1 & {- 1} \\ 1 & {- j} & {- 1} & j \end{bmatrix}.}}$ Also, diag(a, b, c, d) is a 4×4 matrix, and diagonal elements of diag(a, b, c, d) are a, b, c, and d, and all the remaining elements are zero.

Codewords W₁, W₂, W₃, W₄, W₅, and W₆ may be defined as follows:

$\begin{matrix} \begin{matrix} {W_{1} = {\frac{1}{\sqrt{2}}*U_{rot}*\begin{bmatrix} 1 & 1 & 0 & 0 \\ 1 & {- 1} & 0 & 0 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 1 & {- 1} \end{bmatrix}*\begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & {- 1} & 0 \\ 0 & 0 & 0 & {- 1} \end{bmatrix}}} \\ {= {0.5*\begin{bmatrix} 1 & 1 & 1 & 1 \\ 1 & {- 1} & 1 & {- 1} \\ 1 & 1 & {- 1} & {- 1} \\ 1 & {- 1} & {- 1} & 1 \end{bmatrix}}} \end{matrix} & \; \\ \begin{matrix} {W_{2} = {\frac{1}{\sqrt{2}}*U_{rot}*\begin{bmatrix} 1 & 1 & 0 & 0 \\ j & {- j} & 0 & 0 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & j & {- j} \end{bmatrix}*\begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & {- 1} & 0 \\ 0 & 0 & 0 & {- 1} \end{bmatrix}}} \\ {= {0.5*\begin{bmatrix} 1 & 1 & 1 & 1 \\ j & {- j} & j & {- j} \\ 1 & 1 & {- 1} & {- 1} \\ j & {- j} & {- j} & j \end{bmatrix}}} \end{matrix} & \; \\ \begin{matrix} {W_{3} = {\frac{1}{\sqrt{2}}*U_{rot}*\begin{bmatrix} 1 & 1 & 0 & 0 \\ 1 & {- 1} & 0 & 0 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & j & {- j} \end{bmatrix}*\begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & {- 1} & 0 \\ 0 & 0 & 0 & {- 1} \end{bmatrix}}} \\ {= {0.5*\begin{bmatrix} 1 & 1 & 1 & 1 \\ 1 & {- 1} & j & {- j} \\ 1 & 1 & {- 1} & {- 1} \\ 1 & {- 1} & {- j} & j \end{bmatrix}}} \end{matrix} & \; \\ \begin{matrix} {W_{4\;} = {\frac{1}{\sqrt{2}}*U_{rot}*\begin{bmatrix} 1 & 1 & 0 & 0 \\ j & {- j} & 0 & 0 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 1 & {- 1} \end{bmatrix}*\begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & {- 1} & 0 \\ 0 & 0 & 0 & {- 1} \end{bmatrix}}} \\ {= {0.5*\begin{bmatrix} 1 & 1 & 1 & 1 \\ j & {- j} & 1 & {- 1} \\ 1 & 1 & {- 1} & {- 1} \\ j & {- j} & {- 1} & 1 \end{bmatrix}}} \end{matrix} & \; \\ {W_{5} = {{{{diag}\left( {1,1,1,{- 1}} \right)}*{DFT}}\mspace{34mu} = {0.5*\begin{bmatrix} 1 & 1 & 1 & 1 \\ 1 & j & {- 1} & {- j} \\ 1 & {- 1} & 1 & {- 1} \\ {- 1} & j & 1 & {- j} \end{bmatrix}}}} & \; \\ \begin{matrix} {W_{6} = {{{diag}\left( {1,\frac{\left( {1 + j} \right)}{\sqrt{2}},j,\frac{\left( {{- 1} + j} \right)}{\sqrt{2}}} \right)}*{DFT}}} \\ {= {0.5*\begin{bmatrix} 1 & 1 & 1 & 1 \\ \frac{\left( {1 + j} \right)}{\sqrt{2}} & \frac{\left( {{- 1} + j} \right)}{\sqrt{2}} & \frac{\left( {{- 1} - j} \right)}{\sqrt{2}} & \frac{\left( {1 - j} \right)}{\sqrt{2}} \\ j & {- j} & j & {- j} \\ \frac{\left( {{- 1} + j} \right)}{\sqrt{2}} & \frac{\left( {1 + j} \right)}{\sqrt{2}} & \frac{\left( {1 - j} \right)}{\sqrt{2}} & \frac{\left( {{- 1} - j} \right)}{\sqrt{2}} \end{bmatrix}}} \end{matrix} & \; \end{matrix}$

Where the number of physical antennas of the base station is four, the matrices or the codewords included in the codebook used in the single user MIMO communication system according to the fourth example may be given by the following Table 10:

Transmit Transmission Transmission Transmission Transmission Codebook Index Rank 1 Rank 2 Rank 3 Rank 4 1 C_(1,1)=W1(;,2) C_(1,2)=W1(:,12) C_(1,3)=W1(;,1 2 3) C_(1,4)=W1(;,1 2 3 4) 2 C_(2,1)=W1(;,3) C_(2,2)=W1(:,13) C_(2,3)=W1(;,1 2 4) C_(2,4)=W2(;,1 2 3 4) 3 C_(3,1)=W1(;,4) C_(3,2)=W1(:,14) C_(3,3)=W1(;,1 3 4) C_(3,4)=W3(;,1 2 3 4) 4 C_(4,1)=W2(;,2) C_(4,2)=W1(:,23) C_(4,3)=W1(;,2 3 4) C_(4,4)=W4(;,1 2 3 4) 5 C_(5,1)=W2(;,3) C_(5,2)=W1(:,24) C_(5,3)=W2(;,1 2 3) C_(5,4)=W5(;,1 2 3 4) 6 C_(6,1)=W2(;,4) C_(6,2)=W1(:,34) C_(6,3)=W2(;,1 2 4) C_(6,4)=W6(;,1 2 3 4) 7 C_(7,1)=W3(;,1) C_(7,2)=W2(:,13) C_(7,3)=W2(;,1 3 4) n/a 8 C_(8,1)=W4(;,1) C_(8,2)=W2(:,14) C_(8,3)=W2(;,2 3 4) n/a 9 C_(9,1)=W5(;,1) C_(9,2)=W2(:,23) C_(9,3)=W3(;,1 2 3) n/a 10 C_(10,1)=W5(;,2) C_(10,2)=W2(:,24) C_(10,3)=W3(;,1 3 4) n/a 11 C_(11,1)=W5(;,3) C_(11,2)=W3(:,13) C_(11,3)=W4(;,1 2 3) n/a 12 C_(12,1)=W5(;,4) C_(12,2)=W3(:,14) C_(12,3)=W4(;,1 3 4) n/a 13 C_(13,1)=W6(;,1) C_(13,2)=W4(:,13) C_(13,3)=W5(;,1 2 3) n/a 14 C_(14,1)=W6(;,2) C_(14,2)=W4(:,14) C_(14,3)=W5(;,1 3 4) n/a 15 C_(15,1)=W6(;,3) C_(15,2)=W5(:,13) C_(15,3)=W6(;,1 2 4) n/a 16 C_(16,1)=W6(;,4) C_(16,2)=W6(:,24) C_(16,3)=W6(;,2 3 4) n/a

The codewords included in the above Table 10 may be expressed equivalently as follows:

$C_{1,1} = \begin{matrix} 0.5000 \\ {- 0.5000} \\ 0.5000 \\ {- 0.5000} \end{matrix}$ $C_{2,1} = \begin{matrix} 0.5000 \\ 0.5000 \\ {- 0.5000} \\ {- 0.5000} \end{matrix}$ $C_{3,1} = \begin{matrix} 0.5000 \\ {- 0.5000} \\ {- 0.5000} \\ 0.5000 \end{matrix}$ $C_{4,1} = \begin{matrix} 0.5000 \\ {0 - {0.5000{\mathbb{i}}}} \\ 0.5000 \\ {0 - {0.5000{\mathbb{i}}}} \end{matrix}$

$C_{5,1} = \begin{matrix} 0.5000 \\ {0 + {0.5000{\mathbb{i}}}} \\ {- 0.5000} \\ {0 - {0.5000{\mathbb{i}}}} \end{matrix}$ $C_{6,1} = \begin{matrix} 0.5000 \\ {0 - {0.5000{\mathbb{i}}}} \\ {- 0.5000} \\ {0 + {0.5000{\mathbb{i}}}} \end{matrix}$ $C_{7,1} = \begin{matrix} 0.5000 \\ 0.5000 \\ 0.5000 \\ 0.5000 \end{matrix}$ $C_{8,1} = \begin{matrix} 0.5000 \\ {0 + {0.5000{\mathbb{i}}}} \\ 0.5000 \\ {0 + {0.5000{\mathbb{i}}}} \end{matrix}$

$C_{9,1} = \begin{matrix} 0.5000 \\ 0.5000 \\ 0.5000 \\ {- 0.5000} \end{matrix}$ $C_{10,1} = \begin{matrix} 0.5000 \\ {0 + {0.5000{\mathbb{i}}}} \\ {- 0.5000} \\ {0 + {0.5000{\mathbb{i}}}} \end{matrix}$ $C_{11,1} = \begin{matrix} 0.5000 \\ {- 0.5000} \\ 0.5000 \\ 0.5000 \end{matrix}$ $C_{12,1} = \begin{matrix} 0.5000 \\ {0 - {0.5000{\mathbb{i}}}} \\ {- 0.5000} \\ {0 - {0.5000{\mathbb{i}}}} \end{matrix}$

$C_{13,1} = \begin{matrix} 0.5000 \\ {0.3536 + {03536{\mathbb{i}}}} \\ {0 + {0.5000{\mathbb{i}}}} \\ {{- 0.3536} + {0.3536{\mathbb{i}}}} \end{matrix}$ $C_{14,1} = \begin{matrix} 0.5000 \\ {{- 0.3536} + {0.3536{\mathbb{i}}}} \\ {0 - {0.5000{\mathbb{i}}}} \\ {0.3536 + {03536{\mathbb{i}}}} \end{matrix}$ $C_{15,1} = \begin{matrix} 0.5000 \\ {{- 0.3536} - {0.3536{\mathbb{i}}}} \\ {0 + {0.5000{\mathbb{i}}}} \\ {0.3536 - {03536{\mathbb{i}}}} \end{matrix}$ $C_{16,1} = \begin{matrix} 0.5000 \\ {0.3536 - {0.3536{\mathbb{i}}}} \\ {0 - {0.5000{\mathbb{i}}}} \\ {{- 0.3536} - {03536{\mathbb{i}}}} \end{matrix}$

$C_{1,2} = \begin{matrix} 0.5000 & 0.5000 \\ {- 0.5000} & 0.5000 \\ 0.5000 & 0.5000 \\ {- 0.5000} & 0.5000 \end{matrix}$ $C_{2,2} = \begin{matrix} 0.5000 & 0.5000 \\ 0.5000 & 0.5000 \\ {- 0.5000} & 0.5000 \\ {- 0.5000} & 0.5000 \end{matrix}$ $C_{3,2} = \begin{matrix} 0.5000 & 0.5000 \\ {- 0.5000} & 0.5000 \\ {- 0.5000} & 0.5000 \\ 0.5000 & 0.5000 \end{matrix}$ $C_{4,2} = \begin{matrix} 0.5000 & 0.5000 \\ {- 0.5000} & 0.5000 \\ 0.5000 & {- 0.5000} \\ {- 0.5000} & {- 0.5000} \end{matrix}$

$C_{5,2} = \begin{matrix} 0.5000 & 0.5000 \\ {- 0.5000} & {- 0.5000} \\ 0.5000 & {- 0.5000} \\ {- 0.5000} & 0.5000 \end{matrix}$ $C_{6,2} = \begin{matrix} 0.5000 & 0.5000 \\ 0.5000 & {- 0.5000} \\ {- 0.5000} & {- 0.5000} \\ {- 0.5000} & 0.5000 \end{matrix}$ $C_{7,2} = \begin{matrix} 0.5000 & 0.5000 \\ {0 + {0.5000{\mathbb{i}}}} & {0 + {0.5000{\mathbb{i}}}} \\ {- 0.5000} & 0.5000 \\ {0 - {0.5000{\mathbb{i}}}} & {0 + {0.5000{\mathbb{i}}}} \end{matrix}$ $C_{8,2} = \begin{matrix} 0.5000 & 0.5000 \\ {0 - {0.5000{\mathbb{i}}}} & {0 + {0.5000{\mathbb{i}}}} \\ {- 0.5000} & 0.5000 \\ {0 + {0.5000{\mathbb{i}}}} & {0 + {0.5000{\mathbb{i}}}} \end{matrix}$

$C_{9,2} = \begin{matrix} 0.5000 & 0.5000 \\ {0 - {0.5000{\mathbb{i}}}} & {0 + {0.5000{\mathbb{i}}}} \\ 0.5000 & {- 0.5000} \\ {0 - {0.5000{\mathbb{i}}}} & {0 - {0.5000{\mathbb{i}}}} \end{matrix}$ $C_{10,2} = \begin{matrix} 0.5000 & 0.5000 \\ {0 - {0.5000{\mathbb{i}}}} & {0 - {0.5000{\mathbb{i}}}} \\ 0.5000 & {- 0.5000} \\ {0 - {0.5000{\mathbb{i}}}} & {0 + {0.5000{\mathbb{i}}}} \end{matrix}$ $C_{11,2} = \begin{matrix} 0.5000 & 0.5000 \\ {0 + {0.5000{\mathbb{i}}}} & 0.5000 \\ {- 0.5000} & 0.5000 \\ {0 - {0.5000{\mathbb{i}}}} & 0.5000 \end{matrix}$ $C_{12,2} = \begin{matrix} 0.5000 & 0.5000 \\ {0 - {0.5000{\mathbb{i}}}} & 0.5000 \\ {- 0.5000} & 0.5000 \\ {0 + {0.5000{\mathbb{i}}}} & 0.5000 \end{matrix}$

$C_{13,2} = \begin{matrix} 0.5000 & 0.5000 \\ 0.5000 & {0 + {0.5000{\mathbb{i}}}} \\ {- 0.5000} & 0.5000 \\ {- 0.5000} & {0 + {0.5000{\mathbb{i}}}} \end{matrix}$ $C_{14,2} = \begin{matrix} 0.5000 & 0.5000 \\ {- 0.5000} & {0 + {0.5000{\mathbb{i}}}} \\ {- 0.5000} & 0.5000 \\ 0.5000 & {0 + {0.5000{\mathbb{i}}}} \end{matrix}$ $C_{15,2} = \begin{matrix} 0.5000 & 0.5000 \\ 0.5000 & {- 0.5000} \\ 0.5000 & 0.5000 \\ {- 0.5000} & 0.5000 \end{matrix}$ $C_{16,2} = \begin{matrix} 0.5000 & 0.5000 \\ {{- 0.3536} + {0.3536{\mathbb{i}}}} & {0.3536 - {0.3536{\mathbb{i}}}} \\ {0 - {0.5000{\mathbb{i}}}} & {0 - {0.5000{\mathbb{i}}}} \\ {0.3536 + {0.3536{\mathbb{i}}}} & {{- 0.3536} - {0.3536{\mathbb{i}}}} \end{matrix}$

$C_{1,3} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 \\ 0.5000 & {- 0.5000} & 0.5000 \\ 0.5000 & 0.5000 & {- 0.5000} \\ 0.5000 & {- 0.5000} & {- 0.5000} \end{matrix}$ $C_{2,3} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 \\ 0.5000 & {- 0.5000} & {- 0.5000} \\ 0.5000 & 0.5000 & {- 0.5000} \\ 0.5000 & {- 0.5000} & 0.5000 \end{matrix}$ $C_{3,3} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 \\ 0.5000 & 0.5000 & {- 0.5000} \\ 0.5000 & {- 0.5000} & {- 0.5000} \\ 0.5000 & {- 0.5000} & 0.5000 \end{matrix}$ $C_{4,3} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 \\ {- 0.5000} & 0.5000 & {- 0.5000} \\ 0.5000 & {- 0.5000} & {- 0.5000} \\ {- 0.5000} & {- 0.5000} & 0.5000 \end{matrix}$

$C_{5,3} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 \\ {0 + {0.5000{\mathbb{i}}}} & {0 - {0.5000{\mathbb{i}}}} & {0 + {0.5000{\mathbb{i}}}} \\ 0.5000 & 0.5000 & {- 0.5000} \\ {0 + {0.5000{\mathbb{i}}}} & {0 - {0.5000{\mathbb{i}}}} & {0 - {0.5000{\mathbb{i}}}} \end{matrix}$ $C_{6,3} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 \\ {0 + {0.5000{\mathbb{i}}}} & {0 - {0.5000{\mathbb{i}}}} & {0 - {0.5000{\mathbb{i}}}} \\ 0.5000 & 0.5000 & {- 0.5000} \\ {0 + {0.5000{\mathbb{i}}}} & {0 - {0.5000{\mathbb{i}}}} & {0 + {0.5000{\mathbb{i}}}} \end{matrix}$ $C_{7,3} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 \\ {0 + {0.5000{\mathbb{i}}}} & {0 + {0.5000{\mathbb{i}}}} & {0 - {0.5000{\mathbb{i}}}} \\ 0.5000 & {- 0.5000} & {- 0.5000} \\ {0 + {0.5000{\mathbb{i}}}} & {0 - {0.5000{\mathbb{i}}}} & {0 + {0.5000{\mathbb{i}}}} \end{matrix}$ $C_{8,3} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 \\ {0 - {0.5000{\mathbb{i}}}} & {0 + {0.5000{\mathbb{i}}}} & {0 - {0.5000{\mathbb{i}}}} \\ 0.5000 & {- 0.5000} & {- 0.5000} \\ {0 - {0.5000{\mathbb{i}}}} & {0 - {0.5000{\mathbb{i}}}} & {0 + {0.5000{\mathbb{i}}}} \end{matrix}$

$C_{9,3} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 \\ 0.5000 & {- 0.5000} & {0 + {0.5000{\mathbb{i}}}} \\ 0.5000 & 0.5000 & {- 0.5000} \\ 0.5000 & {- 0.5000} & {0 - {0.5000{\mathbb{i}}}} \end{matrix}$ $C_{10,3} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 \\ 0.5000 & {0 + {0.5000{\mathbb{i}}}} & {0 - {0.5000{\mathbb{i}}}} \\ 0.5000 & {- 0.5000} & {- 0.5000} \\ 0.5000 & {0 - {0.5000{\mathbb{i}}}} & {0 + {0.5000{\mathbb{i}}}} \end{matrix}$ $C_{11,3} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 \\ {0 + {0.5000{\mathbb{i}}}} & {0 - {0.5000{\mathbb{i}}}} & 0.5000 \\ 0.5000 & 0.5000 & {- 0.5000} \\ {0 + {0.5000{\mathbb{i}}}} & {0 - {0.5000{\mathbb{i}}}} & {- 0.5000} \end{matrix}$ $C_{12,3} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 \\ {0 + {0.5000{\mathbb{i}}}} & 0.5000 & {- 0.5000} \\ 0.5000 & {- 0.5000} & {- 0.5000} \\ {0 + {0.5000{\mathbb{i}}}} & {- 0.5000} & 0.5000 \end{matrix}$

$C_{13,3} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 \\ 0.5000 & {0 + {0.5000{\mathbb{i}}}} & {- 0.5000} \\ 0.5000 & {- 0.5000} & 0.5000 \\ {- 0.5000} & {0 + {0.5000{\mathbb{i}}}} & 0.5000 \end{matrix}$ $C_{14,3} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 \\ 0.5000 & {- 0.5000} & {0 - {0.5000{\mathbb{i}}}} \\ 0.5000 & 0.5000 & {- 0.5000} \\ {- 0.5000} & 0.5000 & {0 - {0.5000{\mathbb{i}}}} \end{matrix}$ $C_{15,3} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 \\ \begin{matrix} {0.3536 +} \\ {0.3536{\mathbb{i}}} \end{matrix} & \begin{matrix} {{- 0.3536} +} \\ {0.3536{\mathbb{i}}} \end{matrix} & \begin{matrix} {0.3536 -} \\ {0.3536{\mathbb{i}}} \end{matrix} \\ {0 + {0.5000{\mathbb{i}}}} & {0 - {0.5000{\mathbb{i}}}} & {0 - {0.5000{\mathbb{i}}}} \\ \begin{matrix} {{- 0.3536} +} \\ {0.3536{\mathbb{i}}} \end{matrix} & \begin{matrix} {0.3536 +} \\ {0.3536{\mathbb{i}}} \end{matrix} & \begin{matrix} {{- 0.3536} -} \\ {0.3536{\mathbb{i}}} \end{matrix} \end{matrix}$ $C_{16,3} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 \\ \begin{matrix} {{- 0.3536} +} \\ {0.3536{\mathbb{i}}} \end{matrix} & \begin{matrix} {{- 0.3536} -} \\ {0.3536{\mathbb{i}}} \end{matrix} & \begin{matrix} {0.3536 -} \\ {0.3536{\mathbb{i}}} \end{matrix} \\ {0 - {0.5000{\mathbb{i}}}} & {0 + {0.5000{\mathbb{i}}}} & {0 - {0.5000{\mathbb{i}}}} \\ \begin{matrix} {0.3536 +} \\ {0.3536{\mathbb{i}}} \end{matrix} & \begin{matrix} {0.3536 -} \\ {0.3536{\mathbb{i}}} \end{matrix} & \begin{matrix} {{- 0.3536} -} \\ {0.3536{\mathbb{i}}} \end{matrix} \end{matrix}$

$C_{1,4} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 & 0.5000 \\ 0.5000 & {- 0.5000} & 0.5000 & {- 0.5000} \\ 0.5000 & 0.5000 & {- 0.5000} & {- 0.5000} \\ 0.5000 & {- 0.5000} & {- 0.5000} & 0.5000 \end{matrix}$ $C_{2,4} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 & 0.5000 \\ {0 + {0.5000{\mathbb{i}}}} & {0 - {0.5000{\mathbb{i}}}} & {0 + {0.5000{\mathbb{i}}}} & {0 - {0.5000{\mathbb{i}}}} \\ 0.5000 & 0.5000 & {- 0.5000} & {- 0.5000} \\ {0 + {0.5000{\mathbb{i}}}} & {0 - {0.5000{\mathbb{i}}}} & {0 - {0.5000{\mathbb{i}}}} & {0 + {0.5000{\mathbb{i}}}} \end{matrix}$ $C_{3,4} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 & 0.5000 \\ 0.5000 & {- 0.5000} & {0 + {0.5000{\mathbb{i}}}} & {0 - {0.5000{\mathbb{i}}}} \\ 0.5000 & 0.5000 & {- 0.5000} & {- 0.5000} \\ 0.5000 & {- 0.5000} & {0 - {0.5000{\mathbb{i}}}} & {0 + {0.5000{\mathbb{i}}}} \end{matrix}$ $C_{4,4} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 & 0.5000 \\ {0 + {0.5000{\mathbb{i}}}} & {0 - {0.5000{\mathbb{i}}}} & 0.5000 & {- 0.5000} \\ 0.5000 & 0.5000 & {- 0.5000} & {- 0.5000} \\ {0 + {0.5000{\mathbb{i}}}} & {0 - {0.5000{\mathbb{i}}}} & {- 0.5000} & 0.5000 \end{matrix}$

$C_{5,4} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 & 0.5000 \\ 0.5000 & {0 + {0.5000{\mathbb{i}}}} & {- 0.5000} & {0 - {0.5000{\mathbb{i}}}} \\ 0.5000 & {- 0.5000} & 0.5000 & {- 0.5000} \\ {- 0.5000} & {0 + {0.5000{\mathbb{i}}}} & 0.5000 & {0 - {0.5000{\mathbb{i}}}} \end{matrix}$ $C_{6,4} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 & 0.5000 \\ \begin{matrix} {0.3536 +} \\ {0.3536{\mathbb{i}}} \end{matrix} & \begin{matrix} {{- 0.3536} +} \\ {0.3536{\mathbb{i}}} \end{matrix} & \begin{matrix} {{- 0.3536} -} \\ {0.3536{\mathbb{i}}} \end{matrix} & \begin{matrix} {0.3536 -} \\ {0.3536{\mathbb{i}}} \end{matrix} \\ {0 + {0.5000{\mathbb{i}}}} & {0 - {0.5000{\mathbb{i}}}} & {0 + {0.5000{\mathbb{i}}}} & {0 - {0.5000{\mathbb{i}}}} \\ \begin{matrix} {{- 0.3536} +} \\ {0.3536{\mathbb{i}}} \end{matrix} & \begin{matrix} {0.3536 +} \\ {0.3536{\mathbb{i}}} \end{matrix} & \begin{matrix} {0.3536 -} \\ {0.3536{\mathbb{i}}} \end{matrix} & \begin{matrix} {{- 0.3536} -} \\ {0.3536{\mathbb{i}}} \end{matrix} \end{matrix}$

For the fourth example, where the number of physical antennas of the base station is four, the codebook used in the multi-user MIMO communication system according to the fourth example may be designed by combining subsets of two matrices W₃ and W₆ as follows:

$\begin{matrix} \begin{matrix} {W_{3} = {\frac{1}{\sqrt{2}}*U_{rot}*\begin{bmatrix} 1 & 1 & 0 & 0 \\ 1 & {- 1} & 0 & 0 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & j & {- j} \end{bmatrix}*\begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & {- 1} & 0 \\ 0 & 0 & 0 & {- 1} \end{bmatrix}}} \\ {= {0.5*\begin{bmatrix} 1 & 1 & 1 & 1 \\ 1 & {- 1} & j & {- j} \\ 1 & 1 & {- 1} & {- 1} \\ 1 & {- 1} & {- j} & j \end{bmatrix}}} \end{matrix} & \; \\ \begin{matrix} {W_{6} = {{{diag}\left( {1,\frac{\left( {1 + j} \right)}{\sqrt{2}},j,\frac{\left( {{- 1} + j} \right)}{\sqrt{2}}} \right)}*{DFT}}} \\ {= {0.5*\begin{bmatrix} 1 & 1 & 1 & 1 \\ \frac{\left( {1 + j} \right)}{\sqrt{2}} & \frac{\left( {{- 1} + j} \right)}{\sqrt{2}} & \frac{\left( {{- 1} - j} \right)}{\sqrt{2}} & \frac{\left( {1 - j} \right)}{\sqrt{2}} \\ j & {- j} & j & {- j} \\ \frac{\left( {{- 1} + j} \right)}{\sqrt{2}} & \frac{\left( {1 + j} \right)}{\sqrt{2}} & \frac{\left( {1 - j} \right)}{\sqrt{2}} & \frac{\left( {{- 1} - j} \right)}{\sqrt{2}} \end{bmatrix}}} \end{matrix} & \; \end{matrix}$

Where the number of physical antennas of the base station is four, the matrices or the codewords included in the codebook used in the multi-user MIMO communication system according to the fourth example may be expressed using a transmission rank, as given by the following Table 11:

codeword used at Transmit Codebook the mobile station Index for quantization 1 M₁=W3(;,1) 2 M₂=W3(;,2) 3 M₃=W3(;,3) 4 M₄=W3(;,4) 5 M₅=W6(;,1) 6 M₆=W6(;,2) 7 M₇=W6(;,3) 8 M₈=W6(;,4)

The codewords included in the above Table 11 may be expressed equivalently as follows:

$M_{1} = \begin{matrix} 0.5000 \\ 0.5000 \\ 0.5000 \\ 0.5000 \end{matrix}$ $M_{2} = \begin{matrix} 0.5000 \\ {- 0.5000} \\ 0.5000 \\ {- 0.5000} \end{matrix}$ $M_{3} = \begin{matrix} 0.5000 \\ {0 + {0.5000{\mathbb{i}}}} \\ {- 0.5000} \\ {0 - {0.5000{\mathbb{i}}}} \end{matrix}$ $M_{4} = \begin{matrix} 0.5000 \\ {0 - {0.5000{\mathbb{i}}}} \\ {- 0.5000} \\ {0 + {0.5000{\mathbb{i}}}} \end{matrix}$

$M_{5} = \begin{matrix} 0.5000 \\ {0.3536 + {0.3536{\mathbb{i}}}} \\ {0 + {0.5000{\mathbb{i}}}} \\ {{- 0.3536} + {0.3536{\mathbb{i}}}} \end{matrix}$ $M_{6} = \begin{matrix} 0.5000 \\ {{- 0.3536} + {0.3536{\mathbb{i}}}} \\ {0 - {0.5000{\mathbb{i}}}} \\ {0.3536 + {0.3536{\mathbb{i}}}} \end{matrix}$ $M_{7} = \begin{matrix} 0.5000 \\ {{- 0.3536} - {0.3536{\mathbb{i}}}} \\ {0 + {0.5000{\mathbb{i}}}} \\ {0.3536 - {0.3536{\mathbb{i}}}} \end{matrix}$ $M_{8} = \begin{matrix} 0.5000 \\ {0.3536 - {0.3536{\mathbb{i}}}} \\ {0 - {0.5000{\mathbb{i}}}} \\ {{- 0.3536} - {0.3536{\mathbb{i}}}} \end{matrix}$

In this example, the codebook used in the multi-user MIMO communication system performing non-unitary precoding has a base station that includes four physical antennas.

Codewords included in the codebook used in the multi-user MIMO communication system performing non-unitary precoding may be given by the following Table 12:

Transmit Codebook Transmission Index Rank 1 1 C_(1,1)=W1(;,2) 2 C_(2,1)=W1(;,3) 3 C_(3,1)=W1(;,4) 4 C_(4,1)=W2(;,2) 5 C_(5,1)=W2(;,3) 6 C_(6,1)=W2(;,4) 7 C_(7,1)=W3(;,1) 8 C_(8,1)=W4(;,1) 9 C_(9,1)=W5(;,1) 10 C_(10,1)=W5(;,2) 11 C_(11,1)=W5(;,3) 12 C_(12,1)=W5(;,4) 13 C_(13,1)=W6(;,1) 14 C_(14,1)=W6(;,2) 15 C_(15,1)=W6(;,3) 16 C_(16,1)=W6(;,4)

The codewords included in the above Table 12 may be expressed equivalently as follows:

$C_{1,1} = \begin{matrix} 0.5000 \\ {- 0.5000} \\ 0.5000 \\ {- 0.5000} \end{matrix}$ $C_{2,1} = \begin{matrix} 0.5000 \\ 0.5000 \\ {- 0.5000} \\ {- 0.5000} \end{matrix}$ $C_{3,1} = \begin{matrix} 0.5000 \\ {- 0.5000} \\ {- 0.5000} \\ 0.5000 \end{matrix}$ $C_{4,1} = \begin{matrix} 0.5000 \\ {0 - {0.5000{\mathbb{i}}}} \\ 0.5000 \\ {0 - {0.5000{\mathbb{i}}}} \end{matrix}$

$C_{5,1} = \begin{matrix} 0.5000 \\ {0 + {0.5000{\mathbb{i}}}} \\ {- 0.5000} \\ {0 - {0.5000{\mathbb{i}}}} \end{matrix}$ $C_{6,1} = \begin{matrix} 0.5000 \\ {0 - {0.5000{\mathbb{i}}}} \\ {- 0.5000} \\ {0 + {0.5000{\mathbb{i}}}} \end{matrix}$ $C_{7,1} = \begin{matrix} 0.5000 \\ 0.5000 \\ 0.5000 \\ 0.5000 \end{matrix}$ $C_{8,1} = \begin{matrix} 0.5000 \\ {0 + {0.5000{\mathbb{i}}}} \\ 0.5000 \\ {0 + {0.5000{\mathbb{i}}}} \end{matrix}$

$C_{9,1} = \begin{matrix} 0.5000 \\ 0.5000 \\ 0.5000 \\ {- 0.5000} \end{matrix}$ $C_{10,1} = \begin{matrix} 0.5000 \\ {0 + {0.5000{\mathbb{i}}}} \\ {- 0.5000} \\ {0 + {0.5000{\mathbb{i}}}} \end{matrix}$ $C_{11,1} = \begin{matrix} 0.5000 \\ {- 0.5000} \\ 0.5000 \\ 0.5000 \end{matrix}$ $C_{12,1} = \begin{matrix} 0.5000 \\ {0 - {0.5000{\mathbb{i}}}} \\ {- 0.5000} \\ {0 - {0.5000{\mathbb{i}}}} \end{matrix}$

$C_{13,1} = \begin{matrix} 0.5000 \\ {03536 + {0.3536{\mathbb{i}}}} \\ {0 + {0.5000{\mathbb{i}}}} \\ {{- 0.3536} + {0.3536{\mathbb{i}}}} \end{matrix}$ $C_{14,1} = \begin{matrix} 0.5000 \\ {{- 03536} + {0.3536{\mathbb{i}}}} \\ {0 - {0.5000{\mathbb{i}}}} \\ {0.3536 + {0.3536{\mathbb{i}}}} \end{matrix}$ $C_{15,1} = \begin{matrix} 0.5000 \\ {{- 03536} - {0.3536{\mathbb{i}}}} \\ {0 + {0.5000{\mathbb{i}}}} \\ {0.3536 - {0.3536{\mathbb{i}}}} \end{matrix}$ $C_{16,1} = \begin{matrix} 0.5000 \\ {03536 - {0.3536{\mathbb{i}}}} \\ {0 - {0.5000{\mathbb{i}}}} \\ {{- 0.3536} - {0.3536{\mathbb{i}}}} \end{matrix}$

Various examples of codebooks and codewords according to a transmission rank and a number of antennas of a base station in a single user MIMO communication system and a multi-user MIMO communication system, have been described above. The aforementioned codewords may be modified using various types of schemes or shapes, and thus are not limited to the aforementioned examples. A similar codebook, or an equivalent codebook may be obtained by changing the phase of the columns of the codewords, for example, by multiplying the columns of the codewords by a complex exponential. As another example, one may multiply the columns of the codewords by ‘−1’.

In some embodiments, the performance or the properties of the codebook may not change when the phases of the columns of the codewords have been changed. Accordingly, it is understood that a codebook generated by changing the phases of the columns of the codewords may be the same as a codebook including the original codewords prior to the changing of the phases. Also, it is understood that a codebook generated by swapping columns of the original codewords of a codebook may be the same as the codebook including the original codewords prior to the swapping. Specific values that are included in the codebook generated by changing the phases of the columns of the original codewords, or in the codebook generated by swapping the columns of the original codewords, will be omitted herein for conciseness.

A fifth example of a codebook used in a single user MIMO communication system where the number of physical antennas of a base station is eight is shown in the following Equation 7:

$\begin{matrix} {W_{0} = {{\frac{1}{\sqrt{8}}\begin{bmatrix} 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\ 1 & {\mathbb{e}}^{j\;{\pi/4}} & {\mathbb{e}}^{{j\pi}/2} & {\mathbb{e}}^{j\; 3{\pi/4}} & {\mathbb{e}}^{j\pi} & {\mathbb{e}}^{{j5\pi}/4} & {\mathbb{e}}^{{j3\pi}/2} & {\mathbb{e}}^{j\; 7{\pi/4}} \\ 1 & {\mathbb{e}}^{j\;{\pi/2}} & {\mathbb{e}}^{j\pi} & {\mathbb{e}}^{j\; 3{\pi/2}} & 1 & {\mathbb{e}}^{j\;{\pi/2}} & {\mathbb{e}}^{j\pi} & {\mathbb{e}}^{j\; 3{\pi/2}} \\ 1 & {\mathbb{e}}^{j\; 3{\pi/4}} & {\mathbb{e}}^{{j3\pi}/2} & {\mathbb{e}}^{j\;{\pi/4}} & {\mathbb{e}}^{j\;\pi} & {\mathbb{e}}^{j\; 7{\pi/4}} & {\mathbb{e}}^{{j\pi}/2} & {\mathbb{e}}^{{j5\pi}/4} \\ 1 & {\mathbb{e}}^{j\;\pi} & 1 & {\mathbb{e}}^{j\pi} & 1 & {\mathbb{e}}^{j\;\pi} & 1 & {\mathbb{e}}^{j\;\pi} \\ 1 & {\mathbb{e}}^{{j5\pi}/4} & {\mathbb{e}}^{j\;{\pi/2}} & {\mathbb{e}}^{j\; 7{\pi/4}} & {\mathbb{e}}^{j\pi} & {\mathbb{e}}^{j\;{\pi/4}} & {\mathbb{e}}^{j\; 3{\pi/2}} & {\mathbb{e}}^{j\; 3{\pi/4}} \\ 1 & {\mathbb{e}}^{{j6\pi}/4} & {\mathbb{e}}^{j\pi} & {\mathbb{e}}^{j\;{\pi/2}} & 1 & {\mathbb{e}}^{j\; 3{\pi/2}} & {\mathbb{e}}^{j\pi} & {\mathbb{e}}^{j\;{\pi/2}} \\ 1 & {\mathbb{e}}^{{j7\pi}/4} & {\mathbb{e}}^{{j3\pi}/2} & {\mathbb{e}}^{j\; 5{\pi/4}} & {\mathbb{e}}^{j\;\pi} & {\mathbb{e}}^{{j3\pi}/4} & {\mathbb{e}}^{{j\pi}/2} & {\mathbb{e}}^{j\;{\pi/4}} \end{bmatrix}}.}} & (7) \end{matrix}$

Matrices included in the codebook for the single user MIMO communication system may be determined using a transmission rank, as given by the following Table 13:

Transmit Codebook Transmission Transmission Transmission Transmission Index Rank 1 Rank 2 Rank 3 Rank 4 1 C1,1 = W0(;,1) C1,2 = W0(;,1 2) C1,3 = W0(;,1 2 3) C1,4 = W0(;,1 2 3 4) 2 C2,1 = W0(;,2) C2,2 = W0(;,3 4) C2,3 = W0(;,3 4 5) C2,4 = W0(;,3 4 5 6) 3 C3,1 = W0(;,3) C3,2 = W0(;,5 6) C3,3 = W0(;,5 6 7) C3,4 = W0(;,5 6 7 8) 4 C4,1 = W0(;,4) C4,2 = W0(;,7 8) C4,3 = W0(;,7 8 1) C4,4 = W0(;,7 8 1 2) 5 C5,1 = W0(;,5) C5,2 = W0(;,1 3) C5,3 = W0(;,1 3 5) C5,4 = W0(;,1 3 5 7) 6 C6,1 = W0(;,6) C6,2 = W0(;,2 4) C6,3 = W0(;,2 4 6) C6,4 = W0(;,2 4 6 8) 7 C7,1 = W0(;,7) C7,2 = W0(;,5 7) C7,3 = W0(;,5 7 1) C7,4 = W0(;,5 7 1 4) 8 C8,1 = W0(;,8) C8,2 = W0(;,6 8) C8,3 = W0(;,6 8 2) C8,4 = W0(;,6 8 2 3)

Referring to the above Table 13, where the transmission rank is 4, the precoding matrix may be generated based on, for example, W₀(;,1 2 3 4), W₀(;,3 4 5 6), W₀(;,5 6 7 8), W₀(;,7 8 1 2), W₀(;,1 3 5 7), W₀(;,2 4 6 8), W₀(;,5 7 1 4), and W₀(;,6 8 2 3). In this example, W_(k)(;,n m o p) denotes a matrix that includes an n^(th) column vector, an m^(th) column vector, an o^(th) column vector, and a p^(th) column vector of W_(k).

Where the transmission rank is 3, the precoding matrix may be generated based on, for example, W₀(;,1 2 3), W₀(;,3 4 5), W₀(;,5 6 7), W₀(;,7 8 1), W₀(;,1 3 5), W₀(;,2 4 6), W_(o)(;,5 7 1), and W₀(;,6 8 2). In this example, W_(k)(;,n m o) denotes a matrix that includes the n^(th) column vector, the m^(th) column vector, and the o^(th) column vector of W_(k).

Where the transmission rank is 2, the precoding matrix may be generated based on, for example, W₀(;,1 2), W₀(;,3 4), W₀(;,5 6), W₀(;,7 8), W₀(;,1 3), W₀(;,2 4), W₀(;,5 7), and W₀(;,6 8). In this example, W_(k)(;,n m) denotes a matrix that includes the n^(th) column vector and the m^(th) column vector of W_(k).

Where the transmission rank is 1, the precoding matrix may be generated based on, for example, W₀(;,1), W₀(;,2), W₀(;,3), W₀(;,4), W₀(;,5), W₀(;,6), W₀(;,7), and W₀(;,8). In this example, W_(k)(;,n) denotes the n^(th) column vector of W_(k).

Design of a 6-Bit Codebook

According to an increase in a number of elements, for example, matrices or vectors, included in a codebook, a feedback overhead may increase whereas a quantization error may decrease. The above tables disclose 4-bit codebooks, each including a plurality of elements. Hereinafter, 6-bit codebooks, including a plurality of elements, for example, 64 elements, will be discussed.

1. First Scheme to Design a 6-Bit Codebook:

In this example, it is assumed that the number of physical antennas of a base station is four, a single user MIMO communication system uses the 4-bit codebooks disclosed in the above Table 1, and a multi-user MIMO communication system uses the codebooks disclosed in the above Table 2:

(1) Operation 1:

The following 4-bit codebook corresponding to the transmission rank 1 may be obtained from the 4-bit codebook disclosed in the above Table 1. In this example, the 4-bit codebook corresponding to the transmission rank 1, disclosed in the above Table 1, is referred to as a base codebook. The base codebook may be expressed using a language according to a numerical analysis program, for example, MATLAB®, manufactured by MathWorks Inc. of Natick, Mass. The numerical analysis language may be, for example, as follows:

rotation_matrix=1/sqrt(2)*[1,0,−1,0;0,1,0,−1;1,0,1,0;0,1,0,1]; W(:,:,1)=1/sqrt(2)*rotation_matrix*[1,1,0,0;1,−1,0,0;0,0,1,1;0,0,1,−1]; W(:,:,2)=1/sqrt(2)*rotation_matrix*[1,1,0,0;1i,−1i,0,0;0,0,1,1;0,0,1i,−1i]; W(:,:,3)=1/sqrt(2)*rotation_matrix*[1,1,0,0;1,−1,0,0;0,0,1,1;0,0,1i,−1i]; W(:,:,4)=1/sqrt(2)*rotation_matrix*[1,1,0,0;1i,−1i,0,0;0,0,1,1;0,0,1,−1]; DFT=1/sqrt(4)*[1,1,1,1;1,1i,−1,−1i;1,−1,1,−1;1,−1i,−1,1i]; W(:,:,5)=diag([1,1,1,−1])*DFT; W(:,:,6)=diag([1,(1+1i)/sqrt(2),1i,(−1+1i)/sqrt(2)])*DFT; base_cbk(:,1:4)=W(:,[1:4],1); base_cbk(:,5:8)=W(:,[1:4],2); base_cbk(:,9:12)=W(:,[1:4],5); base_cbk(:,13:16)=W(:,[1:4],6);

In this example, ‘rotation_matrix’ denotes a rotation matrix U_(rot), ‘sqrt(x)’ denotes √{square root over (x)}, and ‘DFT’ denotes a QPSK DFT matrix. ‘base_cbk(:,a:b)’ denotes an a^(th) element or vector, through a b^(th) element or vector of the base codebook. ‘W(:,[x:y],k)’ denotes a column x^(th) through a y^(th) element among elements of W_(k). For example, ‘base_cbk(:,1:4)’ denotes first, second, third, and fourth columns of the base codebook. ‘W(:,[1:4],1)’ denotes first, second, third, and fourth columns of W₁.

(2) Operation 2:

A local codebook local_cbk may be defined, for example, as follows:

local_cbk= [0.2778+0.4157i   0.2778+0.4157i   0.2778+0.4157i   0.2778+0.4157i −0.0975+0.4904i −0.4904−0.0975i   0.0975−0.4904i   0.4904+0.0975i −0.0975+0.4904i   0.0975−0.4904i −0.0975+0.4904i   0.0975−0.4904i   0.4904+0.0975i   0.0975−0.4904i −0.4904−0.0975i −0.0975+0.4904i];

(3) Operation 3:

The local codebook local_cbk may be scaled to obtain a localized codebook localized_cbk. An exemplary scaling process is as follows:

r=abs(local_cbk); phase_local_cbk=local_cbk./r; alpha= 0.5960

In this example, abs(local_cbk) denotes a magnitude of the local codebook local_cbk. The localized codebook localized_cbk may be expressed as follows:

localized_cbk= [sqrt(1-alpha{circumflex over ( )}2*(1-(r(1,1)){circumflex over ( )}2))*phase_local_cbk(1,1),sqrt(1- alpha{circumflex over ( )}2*(1-(r(1,2)){circumflex over ( )}2))*phase_local_cbk(1,2),sqrt(1- alpha{circumflex over ( )}2*(1-(r(1,3)){circumflex over ( )}2))*phase_local_cbk(1,3),sqrt(1- alpha{circumflex over ( )}2*(1-(r(1,4)){circumflex over ( )}2))*phase_local_cbk(1,4);... alpha*r(2,1)*phase_local_cbk(2,1),alpha*r(2,2)* phase_local_cbk(2,2),alpha*r(2,3)*phase_local_cb k(2,3),alpha*r(2,4)*phase_local_cbk(2,4);... alpha*r(3,1)*phase_local_cbk(3,1),alpha*r(3,2)* phase_local_cbk(3,2),alpha*r(3,3)*phase_local_cb k(3,3),alpha*r(3,4)*phase_local_cbk(3,4);... alpha*r(4,1)*phase_local_cbk(4,1),alpha*r(4,2)* phase_local_cbk(4,2),alpha*r(4,3)*phase_local_cb k(4,3),alpha*r(4,4)*phase_local_cbk(4,4)];

Where the localized codebook localized_cbk is calculated through the aforementioned process, vectors of the localized codebook localized_cbk may be normalized, for example, through the following exemplary process:

for k=1:size(localized_cbk,2) localized_cbk(:,k)=localized_cbk(:,k)/norm(localized_cbk(:,k)); end

(4) Operation 4:

A final codebook final_cbk may be obtained by rotating the normalized localized codebook localized_cbk around the base codebook, for example, through the following exemplary process:

for k=1:16 [U,S,V]=svd(base_cbk(:,k)); %, where a singular value decomposition (SVD) is performed with respect to the elements of the base codebook. R=U′*V; % U′*V*base_cbk(:,1,k)=[1;0;0;0]. % , where R denotes a rotation matrix that rotates the normalized localized codebook localized_cbk around the base codebook. rotated_localized=R′*localized_cbk(:,1:3);%, where rotated_localized is obtained by rotating localized_cbk, and only first three vectors of localized_cbk are rotated. final_cbk(:,(k−1)*4+1:k*4)=[base_cbk(:,k),rotated_localized]; %, where the base codebook is maintained as a centroid and the base codebook is included in a final 6-bit codebook end;

The final 6-bit codebook final_cbk_opt may be generated as follows through the aforementioned operations 1 through 4. The final 6-bit codebook final_cbk_opt may include 64 column vectors.

final_cbk_opt = Columns  1  through  4 $\begin{matrix} 0.5000 & {0.3260 + {0.6774{\mathbb{i}}}} & {0.1499 + {0.0347{\mathbb{i}}}} & {0.0918 + {0.3270{\mathbb{i}}}} \\ 0.5000 & {0.3254 + {0.1709{\mathbb{i}}}} & {0.5009 + {0.3071{\mathbb{i}}}} & {0.1311 + {0.6387{\mathbb{i}}}} \\ 0.5000 & {0.3254 + {0.1709{\mathbb{i}}}} & {0.1505 + {0.5412{\mathbb{i}}}} & {0.2473 + {0.0541{\mathbb{i}}}} \\ 0.5000 & {{- 0.0250} + {0.4051{\mathbb{i}}}} & {0.1505 + {0.5412{\mathbb{i}}}} & {0.4815 + {0.4045{\mathbb{i}}}} \end{matrix}$ Columns  5  through  8 $\begin{matrix} 0.5000 & {0.0918 + {0.3270{\mathbb{i}}}} & {0.3841 + {0.3851{\mathbb{i}}}} & {0.3260 + {0.6774{\mathbb{i}}}} \\ {- 0.5000} & {{- 0.1311} - {0.6387{\mathbb{i}}}} & {0.0056 - {0.30761{\mathbb{i}}}} & {{- 0.3254} - {0.1709{\mathbb{i}}}} \\ 0.5000 & {0.2473 + {0.0541{\mathbb{i}}}} & {0.2285 + {0.6580{\mathbb{i}}}} & {0.3254 + {0.1709{\mathbb{i}}}} \\ {- 0.5000} & {{- 0.4815} - {0.4045{\mathbb{i}}}} & {{- 0.3448} - {0.0735{\mathbb{i}}}} & {0.0250 - {0.4051{\mathbb{i}}}} \end{matrix}$ Columns  9  through  12 $\begin{matrix} {- 0.5000} & {{- 0.0918} - {0.3270{\mathbb{i}}}} & {{- 0.0337} - {0.6193{\mathbb{i}}}} & {{- 0.4422} - {0.0928{\mathbb{i}}}} \\ {- 0.5000} & {{- 0.2473} - {0.0541{\mathbb{i}}}} & {{- 0.4621} - {0.5019{\mathbb{i}}}} & {{- 0.2479} - {0.5606{\mathbb{i}}}} \\ 0.5000 & {0.1311 + {0.6387{\mathbb{i}}}} & {0.2280 + {0.1515{\mathbb{i}}}} & {0.2479 + {0.5606{\mathbb{i}}}} \\ 0.5000 & {0.4815 + {04045{\mathbb{i}}}} & {0.2280 + {0.1515{\mathbb{i}}}} & {0.0137 + {0.2102{\mathbb{i}}}} \end{matrix}$ Columns  13  through  16 $\begin{matrix} {- 0.5000} & {{- 0.4422} - {0.0928{\mathbb{i}}}} & {{- 0.3841} - {0.3851{\mathbb{i}}}} & {{- 0.0918} - {0.3270{\mathbb{i}}}} \\ 0.5000 & {0.2479 + {0.5606{\mathbb{i}}}} & {{- 0.0056} + {0.3076{\mathbb{i}}}} & {0.2473 + {0.0541{\mathbb{i}}}} \\ 0.5000 & {0.2479 + {0.5606{\mathbb{i}}}} & {0.3448 + {0.0735{\mathbb{i}}}} & {0.1311 + {0.6387{\mathbb{i}}}} \\ {- 0.5000} & {{- 0.0137} - {0.2102{\mathbb{i}}}} & {{- 0.2285} - {0.6580{\mathbb{i}}}} & {{- 0.4815} - {0.4045{\mathbb{i}}}} \end{matrix}$

Columns  17  through  20 $\begin{matrix} 0.5000 & {0.3841 + {0.3851{\mathbb{i}}}} & {0.0918 + {0.3270{\mathbb{i}}}} & {0.337 + {0.6193{\mathbb{i}}}} \\ {0 + {0.5000{\mathbb{i}}}} & {{- 0.3076} - {0.0056{\mathbb{i}}}} & {{- 0.0541} + {0.2473{\mathbb{i}}}} & {{- 0.5019} + {0.4621{\mathbb{i}}}} \\ 0.5000 & {0.3448 + {0.0735{\mathbb{i}}}} & {0.1311 + {0.6387{\mathbb{i}}}} & {0.2280 + {0.1515{\mathbb{i}}}} \\ {0 + {0.5000{\mathbb{i}}}} & {{- 0.6580} + {0.2285{\mathbb{i}}}} & {{- 0.4045} + {0.4815{\mathbb{i}}}} & {{- 0.1515} + {0.2280{\mathbb{i}}}} \end{matrix}$ Columns  21  through  24 $\begin{matrix} 0.5000 & {0.0337 + {0.6193{\mathbb{i}}}} & {0.4422 + {0.0928{\mathbb{i}}}} & {0.3841 + {0.3851{\mathbb{i}}}} \\ {0 - {0.5000{\mathbb{i}}}} & {0.5019 - {0.4621{\mathbb{i}}}} & {0.5606 - {0.2479{\mathbb{i}}}} & {0.3076 + {0.0056{\mathbb{i}}}} \\ 0.5000 & {0.2280 + {0.1515{\mathbb{i}}}} & {0.2479 + {0.5606{\mathbb{i}}}} & {0.3448 + {0.0735{\mathbb{i}}}} \\ {0 - {0.5000{\mathbb{i}}}} & {0.1515 - {0.2280{\mathbb{i}}}} & {0.2102 - {0.0137{\mathbb{i}}}} & {0.6580 - {0.2285{\mathbb{i}}}} \end{matrix}$ Columns  25  through  28 $\begin{matrix} {- 0.5000} & {{- 0.3841} - {0.3851{\mathbb{i}}}} & {{- 0.3260} - {0.6774{\mathbb{i}}}} & {{- 0.1499} - {0.0347{\mathbb{i}}}} \\ {0 - {0.5000{\mathbb{i}}}} & {0.3076 + {0.0056{\mathbb{i}}}} & {0.1709 - {0.3254{\mathbb{i}}}} & {0.3071 - {0.5009{\mathbb{i}}}} \\ 0.5000 & {0.2285 + {0.6580{\mathbb{i}}}} & {0.3254 + {0.1709{\mathbb{i}}}} & {0.1505 + {0.5412{\mathbb{i}}}} \\ {0 + {0.5000{\mathbb{i}}}} & {{- 0.0735} + {0.3448{\mathbb{i}}}} & {{- 0.4051} - {0.0250{\mathbb{i}}}} & {{- 0.5412} + {0.1505{\mathbb{i}}}} \end{matrix}$ Columns  29  through  32 $\begin{matrix} {- 0.5000} & {{- 0.1499} - {0.0347{\mathbb{i}}}} & {{- 0.0918} - {0.3270{\mathbb{i}}}} & {{- 0.3841} - {0.3851{\mathbb{i}}}} \\ {0 + {0.5000{\mathbb{i}}}} & {{- 0.3071} + {0.5009{\mathbb{i}}}} & {{- 0.6387} + {0.1311{\mathbb{i}}}} & {{- 0.3076} - {0.0056{\mathbb{i}}}} \\ 0.5000 & {0.1505 + {0.5412{\mathbb{i}}}} & {0.2473 + {0.0541{\mathbb{i}}}} & {0.2285 + {0.6580{\mathbb{i}}}} \\ {0 - {0.5000{\mathbb{i}}}} & {0.5412 - {0.1505{\mathbb{i}}}} & {0.4045 - {0.4815{\mathbb{i}}}} & {0.0735 - {0.3448{\mathbb{i}}}} \end{matrix}$

Columns  33  through  36 $\begin{matrix} 0.5000 & {0.0337 + {0.6193{\mathbb{i}}}} & {0.0918 + {0.3270{\mathbb{i}}}} & {0.3841 + {0.3851{\mathbb{i}}}} \\ 0.5000 & {0.2280 + {0.1515{\mathbb{i}}}} & {0.4815 + {0.4045{\mathbb{i}}}} & {0.2285 + {0.6580{\mathbb{i}}}} \\ 0.5000 & {0.2280 + {0.1515{\mathbb{i}}}} & {0.1311 + {0.6387{\mathbb{i}}}} & {0.3448 + {0.0735{\mathbb{i}}}} \\ {- 0.5000} & {{- 0.4621} - {05019{\mathbb{i}}}} & {{- 0.2473} - {0.0541{\mathbb{i}}}} & {0.0056 - {0.3076{\mathbb{i}}}} \end{matrix}$ Columns  37  through  40 $\begin{matrix} 0.5000 & {0.4422 + {0.0928{\mathbb{i}}}} & {0.0337 + {0.6193{\mathbb{i}}}} & {0.0918 + {0.3270{\mathbb{i}}}} \\ {0 + {0.5000{\mathbb{i}}}} & {{- 0.2102} + {0.0137{\mathbb{i}}}} & {{- 0.1515} + {0.2280{\mathbb{i}}}} & {{- 0.4045} + {0.4815{\mathbb{i}}}} \\ {- 0.5000} & {{- 0.2479} - {0.5606{\mathbb{i}}}} & {{- 0.2280} - {0.1515{\mathbb{i}}}} & {{- 0.1311} - {06387{\mathbb{i}}}} \\ {0 + {0.5000{\mathbb{i}}}} & {{- 0.5606} + {0.2479{\mathbb{i}}}} & {{- 0.5019} + {0.4621{\mathbb{i}}}} & {{- 0.0541} + {0.2473{\mathbb{i}}}} \end{matrix}$ Columns  41  through  44 $\begin{matrix} 0.5000 & {0.3841 + {0.3851{\mathbb{i}}}} & {0.4422 + {0.0928{\mathbb{i}}}} & {0.0337 + {0.6193{\mathbb{i}}}} \\ {- 0.5000} & {{- 0.2285} - {0.6580{\mathbb{i}}}} & {{- 0.0137} - {0.2102{\mathbb{i}}}} & {{- 0.2280} - {01515{\mathbb{i}}}} \\ 0.5000 & {0.3448 + {0.0735{\mathbb{i}}}} & {0.2479 + {0.5606{\mathbb{i}}}} & {0.2280 + {0.1515{\mathbb{i}}}} \\ 0.5000 & {{- 0.0056} + {0.3076{\mathbb{i}}}} & {0.2479 + {0.5606{\mathbb{i}}}} & {0.4621 + {0.5019{\mathbb{i}}}} \end{matrix}$ Columns  45  through  48 $\begin{matrix} 0.5000 & {0.0918 + {0.3270{\mathbb{i}}}} & {0.3841 + {0.3851{\mathbb{i}}}} & {0.4422 + {0.0928{\mathbb{i}}}} \\ {0 - {0.5000{\mathbb{i}}}} & {0.4045 - {0.4815{\mathbb{i}}}} & {0.6580 - {0.2285{\mathbb{i}}}} & {0.2102 - {0.0137{\mathbb{i}}}} \\ {- 0.5000} & {{- 0.1311} - {0.6387{\mathbb{i}}}} & {{- 0.3448} - {0.0735{\mathbb{i}}}} & {{- 0.2479} - {0.5606{\mathbb{i}}}} \\ {0 - {0.5000{\mathbb{i}}}} & {0.0541 - {0.2473{\mathbb{i}}}} & {0.3076 + {0.0056{\mathbb{i}}}} & {0.5606 - {0.2479{\mathbb{i}}}} \end{matrix}$

Columns  49  through  52 $\begin{matrix} 0.5000 & {0.3841 + {0.3851{\mathbb{i}}}} & {{- 0.1560} + {0.4926{\mathbb{i}}}} & {0.3841 + {0.3851{\mathbb{i}}}} \\ {0.3536 + {0.3536{\mathbb{i}}}} & {0.0022 + {0.1690{\mathbb{i}}}} & {0.0837 + {0.4175{\mathbb{i}}}} & {{- 0.1140} + {0.7536{\mathbb{i}}}} \\ {0 + {0.5000{\mathbb{i}}}} & {{- 0.3076} - {0.0056{\mathbb{i}}}} & {{- 0.4597} + {0.3989{\mathbb{i}}}} & {{- 0.3076} - {0.0056{\mathbb{i}}}} \\ {{- 0.3536} + {0.3536{\mathbb{i}}}} & {{- 0.7536} - {0.1140{\mathbb{i}}}} & {{- 0.4175} + {0.0837{\mathbb{i}}}} & {{- 0.1690} + {0.0022{\mathbb{i}}}} \end{matrix}$ Columns  53  through  56 $\begin{matrix} 0.5000 & {0.3396 + {0.1614{\mathbb{i}}}} & {0.3841 + {0.3851{\mathbb{i}}}} & {{- 0.1560} + {0.4926{\mathbb{i}}}} \\ {{- 0.3536} + {0.3536{\mathbb{i}}}} & {{- 03400} - {0.3060{\mathbb{i}}}} & {{- 0.1690} + {0.0022{\mathbb{i}}}} & {{- 0.4175} + {0.0837{\mathbb{i}}}} \\ {0 - {0.5000{\mathbb{i}}}} & {0.3493 - {0.5641{\mathbb{i}}}} & {0.3076 + {0.0056{\mathbb{i}}}} & {0.4597 - {0.3989{\mathbb{i}}}} \\ {0.3536 + {0.3536{\mathbb{i}}}} & {{- 0.3060} + {0.3400{\mathbb{i}}}} & {{- 0.1140} + {0.7536{\mathbb{i}}}} & {0.0837 + {0.4175{\mathbb{i}}}} \end{matrix}$ Columns  57  through  60 $\begin{matrix} 0.5000 & {0.3841 + {0.3851{\mathbb{i}}}} & {0.3396 + {0.1614{\mathbb{i}}}} & {0.3841 + {0.3851{\mathbb{i}}}} \\ {{- 0.3536} - {0.3536{\mathbb{i}}}} & {0.1140 - {0.7536{\mathbb{i}}}} & {0.3060 - {0.3400{\mathbb{i}}}} & {{- 0.0022} - {0.1690{\mathbb{i}}}} \\ {0 + {0.5000{\mathbb{i}}}} & {{- 0.3076} - {0.0056{\mathbb{i}}}} & {{- 0.3493} + {0.5641{\mathbb{i}}}} & {{- 0.3076} - {0.0056{\mathbb{i}}}} \\ {0.3536 - {0.3536{\mathbb{i}}}} & {0.1690 - {0.0022{\mathbb{i}}}} & {0.3400 + {0.3060{\mathbb{i}}}} & {0.7536 + {0.1140{\mathbb{i}}}} \end{matrix}$ Columns  61  through  64 $\begin{matrix} 0.5000 & {{- 0.1560} + {0.4926{\mathbb{i}}}} & {03841 + {0.3851{\mathbb{i}}}} & {0.3396 + {0.1614{\mathbb{i}}}} \\ {0.3536 - {0.3536{\mathbb{i}}}} & {0.4175 - {0.0837{\mathbb{i}}}} & {0.7536 + {0.1140{\mathbb{i}}}} & {0.3400 + {0.3060{\mathbb{i}}}} \\ {0 - {0.5000{\mathbb{i}}}} & {0.4597 - {0.3989{\mathbb{i}}}} & {0.3076 + {0.0056{\mathbb{i}}}} & {0.3493 - {0.5641{\mathbb{i}}}} \\ {{- 0.3536} - {0.3536{\mathbb{i}}}} & {{- 0.0837} - {0.4175{\mathbb{i}}}} & {{- 0.0022} - {0.1690{\mathbb{i}}}} & {0.3060 - {0.3400{\mathbb{i}}}} \end{matrix}$

2. Second Scheme to Design a 6-Bit Codebook:

(1) Operation 1:

Like the first scheme to design the 6-bit codebook, a base codebook may be designed, for example, as follows:

rotation_matrix=1/sqrt(2)*[1,0,−1,0;0,1,0,−1;1,0,1,0;0,1,0,1]; W(:,:,1)=1/sqrt(2)*rotation_matrix*[1,1,0,0;1,−1,0,0;0,0,1,1;0,0,1,−1]; W(:,:,2)=1/sqrt(2)*rotation_matrix*[1,1,0,0;1i,−1i,0,0;0,0,1,1;0,0,1i,−1i]; W(:,:,3)=1/sqrt(2)*rotation_matrix*[1,1,0,0;1,−1,0,0;0,0,1,1;0,0,1i,−1i]; W(:,:,4)=1/sqrt(2)*rotation_matrix*[1,1,0,0;1i,−1i,0,0;0,0,1,1;0,0,1,−1]; DFT=1/sqrt(4)*[1,1,1,1;1,1i,−1,−1i;1,−1,1,−1;1,−1i,−1,1i]; W(:,:,5)=diag([1,1,1,−1])*DFT; W(:,:,6)=diag([1,(1+1i)/sqrt(2),1i,(−1+1i)/sqrt(2)])*DFT; base_cbk(:,1:4)=W(:,[1:4],1); base_cbk(:,5:8)=W(:,[1:4],2); base_cbk(:,9:12)=W(:,[1:4],5); base_cbk(:,13:16)=W(:,[1:4],6);

(2) Operation 2:

A local codebook local_cbk may be defined, for example, as follows:

${{local\_ cbk} = \begin{bmatrix} \begin{matrix} {0.2778 +} \\ {0.4157{\mathbb{i}}} \end{matrix} & \begin{matrix} {0.2778 +} \\ {0.4157{\mathbb{i}}} \end{matrix} & \begin{matrix} {0.2778 +} \\ {0.4157{\mathbb{i}}} \end{matrix} & \begin{matrix} {0.2778 +} \\ {0.4157{\mathbb{i}}} \end{matrix} \\ \begin{matrix} {0.0000 +} \\ {0.5000{\mathbb{i}}} \end{matrix} & \begin{matrix} {{- 0.5000} +} \\ {0.0000{\mathbb{i}}} \end{matrix} & \begin{matrix} {{- 0.0000} -} \\ {0.5000{\mathbb{i}}} \end{matrix} & \begin{matrix} {0.5000 -} \\ {0.0000{\mathbb{i}}} \end{matrix} \\ \begin{matrix} {{- 0.2778} +} \\ {0.4157{\mathbb{i}}} \end{matrix} & \begin{matrix} {0.2778 -} \\ {0.4157{\mathbb{i}}} \end{matrix} & \begin{matrix} {{- 0.2778} +} \\ {0.4157{\mathbb{i}}} \end{matrix} & \begin{matrix} {0.2778 -} \\ {0.4157{\mathbb{i}}} \end{matrix} \\ 0.5000 & {0 - {0.5000{\mathbb{i}}}} & {- 0.5000} & {0 + {0.5000{\mathbb{i}}}} \end{bmatrix}};$

(3) Operation 3:

The local codebook local_cbk may be scaled to obtain a localized codebook localized_cbk. An exemplary scaling process is as follows:

r=abs(local_cbk); phase_local_cbk=local_cbk./r; alpha=0.5716

The localized codebook localized_cbk may be expressed, for example, as follows:

localized_cbk= [sqrt(1−alpha{circumflex over ( )}2*(1−(r(1,1)){circumflex over ( )}2))*phase_local_cbk(1,1),sqrt(1− alpha{circumflex over ( )}2*(1−(r(1,2)){circumflex over ( )}2))*phase_local_cbk(1,2),sqrt(1− alpha{circumflex over ( )}2*(1−(r(1,3)){circumflex over ( )}2))*phase_local_cbk(1,3),sqrt(1− alpha{circumflex over ( )}2*(1−(r(1,4)){circumflex over ( )}2))*phase_local_cbk(1,4);... alpha*r(2,1)*phase_local_cbk(2,1),alpha*r(2,2)* phase_local_cbk(2,2),alpha*r(2,3)*phase_local_cb k(2,3),alpha*r(2,4)*phase_local_cbk(2,4);... alpha*r(3,1)*phase_local_cbk(3,1),alpha*r(3,2)* phase_local_cbk(3,2),alpha*r(3,3)*phase_local_cb k(3,3),alpha*r(3,4)*phase_local_cbk(3,4);... alpha*r(4,1)*phase_local_cbk(4,1),alpha*r(4,2)* phase_local_cbk(4,2),alpha*r(4,3)*phase_local_cb k(4,3),alpha*r(4,4)*phase_local_cbk(4,4)];

Where the localized codebook localized_cbk is calculated through the aforementioned process, vectors of the localized codebook localized_cbk may be normalized, for example, through the following exemplary process:

for k=1:size(localized_cbk,2) localized_cbk(:,k)=localized_cbk(:,k)/norm(localized_cbk(:,k)); end

(4) Operation 4:

A final codebook final_cbk may be obtained by rotating the normalized localized codebook localized_cbk around the base codebook, for example, through the following exemplary process:

  R=U′*V; % U′*V*base_cbk(:,1,k)=[1;0;0;0]. %, where R denotes a rotation matrix that rotates the normalized localized codebook localized_cbk around the base codebook.   rotated_localized=R'*localized_cbk(:,1:3);%, where rotated_localized is obtained by rotating localized_cbk, and only first three vectors of localized_cbk are rotated.   final_cbk(:,(k−1)*4+1:k*4)=[base_cbk(:,k),rotated_localized]; %, where the base codebook is maintained as a centroid and the base codebook is included in a final 6-bit codebook.     end;

The final 6-bit codebook final_cbk_opt may be generated as follows through the aforementioned operations 1 through 4. The final 6-bit codebook final_cbk_opt may include 64 column vectors.

final_cbk_opt = Columns  1  through  4 $\begin{matrix} 0.5000 & {0.3049 + {0.6229{\mathbb{i}}}} & {0.1779 + {0.0995{\mathbb{i}}}} & {0.0919 + {0.3371{\mathbb{i}}}} \\ 0.5000 & {0.2626 + {0.1626{\mathbb{i}}}} & {0.5060 + {0.2740{\mathbb{i}}}} & {0.1673 + {0.6390{\mathbb{i}}}} \\ 0.5000 & {0.4213 + {0.2108{\mathbb{i}}}} & {0.0614 + {0.5116{\mathbb{i}}}} & {0.3261 + {0.1156{\mathbb{i}}}} \\ 0.5000 & {{- 0.0232} + {0.4484{\mathbb{i}}}} & {0.2202 + {0.5598{\mathbb{i}}}} & {0.4531 + {0.3532{\mathbb{i}}}} \end{matrix}$ Columns  5  through  8 $\begin{matrix} 0.5000 & {0.0919 + {03371{\mathbb{i}}}} & {0.4637 + {0.3853{\mathbb{i}}}} & {0.3049 + {0.6229{\mathbb{i}}}} \\ {- 0.5000} & {{- 0.1673} - {06390{\mathbb{i}}}} & {{- 0.0297} - {03692{\mathbb{i}}}} & {{- 0.2626} - {0.1626{\mathbb{i}}}} \\ 0.5000 & {0.3261 + {0.1156{\mathbb{i}}}} & {0.1567 + {0.6069{\mathbb{i}}}} & {0.4213 + {0.2108{\mathbb{i}}}} \\ {- 0.5000} & {{- 0.4531} - {0.3532{\mathbb{i}}}} & {{- 0.3155} - {0.0834{\mathbb{i}}}} & {0.0232 - {0.4484{\mathbb{i}}}} \end{matrix}$ Columns  9  through  12 $\begin{matrix} {- 0.5000} & {{- 0.1779} - {0.3853{\mathbb{i}}}} & {{- 0.0191} - {0.6229{\mathbb{i}}}} & {{- 0.4637} - {0.0995{\mathbb{i}}}} \\ {- 0.5000} & {{- 0.2202} - {0.0834{\mathbb{i}}}} & {{- 0.4531} - {0.4484{\mathbb{i}}}} & {{- 0.3155} - {0.5598{\mathbb{i}}}} \\ 0.5000 & {0.0614 + {0.6069{\mathbb{i}}}} & {0.3261 + {0.2108{\mathbb{i}}}} & {0.1567 + {0.5116{\mathbb{i}}}} \\ 0.5000 & {0.5060 + {0.3692{\mathbb{i}}}} & {0.1673 + {0.1626{\mathbb{i}}}} & {0.0297 + {0.2740{\mathbb{i}}}} \end{matrix}$ Columns  13  through  16 $\begin{matrix} {- 0.5000} & {{- 0.4637} - {0.0995{\mathbb{i}}}} & {{- 0.3049} - {03371{\mathbb{i}}}} & {{- 0.1779} - {03853{\mathbb{i}}}} \\ 0.5000 & {0.3155 + {0.5598{\mathbb{i}}}} & {{- 0.0232} + {0.3532{\mathbb{i}}}} & {0.2202 + {0.0834{\mathbb{i}}}} \\ 0.5000 & {0.1567 + {0.5116{\mathbb{i}}}} & {0.4213 + {0.1156{\mathbb{i}}}} & {0.0614 + {0.6069{\mathbb{i}}}} \\ {- 0.5000} & {{- 0.0297} - {0.2740{\mathbb{i}}}} & {{- 0.2626} - {0.6390{\mathbb{i}}}} & {{- 0.5060} - {0.3692{\mathbb{i}}}} \end{matrix}$

Columns  17  through  20 $\begin{matrix} 0.5000 & {0.3049 + {0.3371{\mathbb{i}}}} & {0.1779 + {0.3853{\mathbb{i}}}} & {0.0191 + {06229{\mathbb{i}}}} \\ {0 + {0.5000{\mathbb{i}}}} & {{- 0.3532} - {0.0232{\mathbb{i}}}} & {{- 0.0834} + {0.2202{\mathbb{i}}}} & {{- 0.4484} + {0.4531{\mathbb{i}}}} \\ 0.5000 & {0.4213 + {0.1156{\mathbb{i}}}} & {0.0614 + {0.6069{\mathbb{i}}}} & {0.3261 + {0.2108{\mathbb{i}}}} \\ {0 + {0.5000{\mathbb{i}}}} & {{- 0.6390} + {0.2626{\mathbb{i}}}} & {{- 0.3692} + {0.5060{\mathbb{i}}}} & {{- 0.1626} + {0.1673{\mathbb{i}}}} \end{matrix}$ Columns  21  through  24 $\begin{matrix} 0.5000 & {0.0191 + {0.6229{\mathbb{i}}}} & {0.4637 + {0.0995{\mathbb{i}}}} & {0.3049 + {0.3371{\mathbb{i}}}} \\ {0 - {0.5000{\mathbb{i}}}} & {0.4484 - {0.4531{\mathbb{i}}}} & {0.5598 - {0.3155{\mathbb{i}}}} & {0.3532 + {0.0232{\mathbb{i}}}} \\ 0.5000 & {0.3261 + {0.2108{\mathbb{i}}}} & {0.1567 + {0.5116{\mathbb{i}}}} & {0.4213 + {0.1156{\mathbb{i}}}} \\ {0 - {0.5000{\mathbb{i}}}} & {0.1626 - {0.1673{\mathbb{i}}}} & {0.2740 - {0.0297{\mathbb{i}}}} & {0.6390 - {0.2626{\mathbb{i}}}} \end{matrix}$ Columns  25  through  28 $\begin{matrix} {- 0.5000} & {{- 0.4637} - {0.3853{\mathbb{i}}}} & {{- 0.3049} - {0.6229{\mathbb{i}}}} & {{- 0.1779} - {0.0995{\mathbb{i}}}} \\ {0 - {0.5000{\mathbb{i}}}} & {0.3692 - {0.0297{\mathbb{i}}}} & {0.1626 - {0.2626{\mathbb{i}}}} & {0.2740 - {0.5060{\mathbb{i}}}} \\ 0.5000 & {0.1567 + {0.6069{\mathbb{i}}}} & {0.4213 + {0.2108{\mathbb{i}}}} & {0.0614 + {0.5116{\mathbb{i}}}} \\ {0 + {0.5000{\mathbb{i}}}} & {{- 0.0834} + {0.3155{\mathbb{i}}}} & {{- 0.4484} - {0.0232{\mathbb{i}}}} & {{- 0.5598} + {0.2202{\mathbb{i}}}} \end{matrix}$ Columns  29  through  32 $\begin{matrix} {- 0.5000} & {{- 0.1779} - {0.0995{\mathbb{i}}}} & {{- 0.0191} - {0.3371{\mathbb{i}}}} & {{- 0.4637} - {0.3853{\mathbb{i}}}} \\ {0 + {0.5000{\mathbb{i}}}} & {{- 0.2740} + {0.5060{\mathbb{i}}}} & {{- 0.6390} + {0.1673{\mathbb{i}}}} & {{- 0.3692} + {0.0297{\mathbb{i}}}} \\ 0.5000 & {0.0614 + {0.5116{\mathbb{i}}}} & {0.3261 + {0.1156{\mathbb{i}}}} & {0.1567 + {0.6069{\mathbb{i}}}} \\ {0 - {0.5000{\mathbb{i}}}} & {0.5598 - {0.2202{\mathbb{i}}}} & {0.3532 - {0.4531{\mathbb{i}}}} & {0.0834 - {03155{\mathbb{i}}}} \end{matrix}$

Columns  33  through  36 $\begin{matrix} 0.5000 & {0.0191 + {0.6229{\mathbb{i}}}} & {0.1779 + {0.3853{\mathbb{i}}}} & {0.3049 + {0.3371{\mathbb{i}}}} \\ 0.5000 & {0.1673 + {0.1626{\mathbb{i}}}} & {0.5060 + {0.3692{\mathbb{i}}}} & {0.2626 + {0.6390{\mathbb{i}}}} \\ 0.5000 & {0.3261 + {0.2108{\mathbb{i}}}} & {0.0614 + {0.6069{\mathbb{i}}}} & {0.4213 + {0.1156{\mathbb{i}}}} \\ {- 0.5000} & {{- 0.4531} - {0.4484{\mathbb{i}}}} & {{- 0.2202} - {0.0834{\mathbb{i}}}} & {0.0232 - {0.3532{\mathbb{i}}}} \end{matrix}$ Columns  37  through  40 $\begin{matrix} 0.5000 & {0.4637 + {0.0995{\mathbb{i}}}} & {0.0191 + {0.6229{\mathbb{i}}}} & {0.1779 + {0.3853{\mathbb{i}}}} \\ {0 + {0.5000{\mathbb{i}}}} & {{- 0.2740} + {0.0297{\mathbb{i}}}} & {{- 0.1626} + {0.1673{\mathbb{i}}}} & {{- 0.3692} + {0.5060{\mathbb{i}}}} \\ {- 0.5000} & {{- 0.1567} - {0.5116{\mathbb{i}}}} & {{- 0.3261} - {0.2108{\mathbb{i}}}} & {{- 0.0614} - {0.6069{\mathbb{i}}}} \\ {0 + {0.5000{\mathbb{i}}}} & {{- 0.5598} + {0.3155{\mathbb{i}}}} & {{- 0.4484} + {0.4531{\mathbb{i}}}} & {{- 0.0834} + {0.2202{\mathbb{i}}}} \end{matrix}$ Columns  41  through  44 $\begin{matrix} 0.5000 & {0.3049 + {0.3371{\mathbb{i}}}} & {0.4637 + {0.0995{\mathbb{i}}}} & {0.0191 + {0.6229{\mathbb{i}}}} \\ {- 0.5000} & {{- 0.2626} - {0.6390{\mathbb{i}}}} & {{- 0.0297} - {0.2740{\mathbb{i}}}} & {{- 0.1673} - {0.1626{\mathbb{i}}}} \\ 0.5000 & {0.4213 + {0.1156{\mathbb{i}}}} & {0.1567 + {0.5116{\mathbb{i}}}} & {0.3261 + {0.2108{\mathbb{i}}}} \\ 0.5000 & {{- 0.0232} + {0.3532{\mathbb{i}}}} & {0.3155 + {0.5598{\mathbb{i}}}} & {0.4531 + {0.4484{\mathbb{i}}}} \end{matrix}$ Columns  45  through  48 $\begin{matrix} 0.5000 & {0.1779 + {0.3853{\mathbb{i}}}} & {0.3049 + {0.3371{\mathbb{i}}}} & {0.4637 + {0.0995{\mathbb{i}}}} \\ {0 - {0.5000{\mathbb{i}}}} & {0.3692 - {0.5060{\mathbb{i}}}} & {0.6390 - {0.2626{\mathbb{i}}}} & {0.2740 - {0.0297{\mathbb{i}}}} \\ {- 0.5000} & {{- 0.0614} - {0.6069{\mathbb{i}}}} & {{- 0.4213} - {0.1156{\mathbb{i}}}} & {{- 0.1567} - {0.5116{\mathbb{i}}}} \\ {0 - {0.5000{\mathbb{i}}}} & {0.0834 - {0.2202{\mathbb{i}}}} & {0.3532 + {0.0232{\mathbb{i}}}} & {0.5598 - {0.3155{\mathbb{i}}}} \end{matrix}$

Columns  49  through  52 $\begin{matrix} 0.5000 & {0.3602 + {0.4406{\mathbb{i}}}} & {{- 0.0795} + {0.4839{\mathbb{i}}}} & {0.3602 + {0.4406{\mathbb{i}}}} \\ {0.03536 + {0.3536{\mathbb{i}}}} & {{- 0.0754} + {0.1870{\mathbb{i}}}} & {0.0965 + {0.3794{\mathbb{i}}}} & {{- 0.0754} + {0.7586{\mathbb{i}}}} \\ {0 + {0.5000\iota}} & {{- 0.2289} + {0.0434{\mathbb{i}}}} & {{- 0.5609} + {0.3720{\mathbb{i}}}} & {{- 0.2289} + {0.0434{\mathbb{i}}}} \\ {{- 0.3536} + {0.3536{\mathbb{i}}}} & {{- 0.7586} - {0.0754{\mathbb{i}}}} & {{- 0.3794} + {0.0965{\mathbb{i}}}} & {{- 0.1870} - {0.0754{\mathbb{i}}}} \end{matrix}$ Columns  53  through  56 $\begin{matrix} 0.5000 & {0.3247 + {0.0797{\mathbb{i}}}} & {0.3602 + {0.4406{\mathbb{i}}}} & {{- 0.0795} + {0.4839{\mathbb{i}}}} \\ {{- 0.03536} + {0.3536{\mathbb{i}}}} & {{- 0.3794} - {0.2846{\mathbb{i}}}} & {{- 0.1870} - {0.0754{\mathbb{i}}}} & {{- 0.3794} + {0.0965{\mathbb{i}}}} \\ {0 - {0.5000{\mathbb{i}}}} & {0.4262 - {0.5068{\mathbb{i}}}} & {0.2289 - {0.0434{\mathbb{i}}}} & {0.5609 - {0.3720{\mathbb{i}}}} \\ {0.3536 + {0.3536{\mathbb{i}}}} & {{- 0.2846} + {0.3794{\mathbb{i}}}} & {{- 0.0754} + {0.7586{\mathbb{i}}}} & {0.0965 + {0.3794{\mathbb{i}}}} \end{matrix}$ Columns  57  through  60 $\begin{matrix} 0.5000 & {0.3602 + {0.4406{\mathbb{i}}}} & {0.3247 + {0.0797{\mathbb{i}}}} & {0.3602 + {0.4406{\mathbb{i}}}} \\ {{- 0.03536} - {0.3536{\mathbb{i}}}} & {0.0754 - {0.7586{\mathbb{i}}}} & {0.2846 - {0.3794{\mathbb{i}}}} & {0.0754 - {0.1870{\mathbb{i}}}} \\ {0 + {0.5000{\mathbb{i}}}} & {{- 0.2289} + {0.0434{\mathbb{i}}}} & {{- 0.4262} + {0.5068{\mathbb{i}}}} & {{- 0.2289} + {0.0434{\mathbb{i}}}} \\ {0.3536 - {0.3536{\mathbb{i}}}} & {0.1870 + {0.0754{\mathbb{i}}}} & {0.3794 + {0.2846{\mathbb{i}}}} & {0.7586 + {0.0754{\mathbb{i}}}} \end{matrix}$ Columns  61  through  64 $\begin{matrix} 0.5000 & {{- 0.0795} + {0.4839{\mathbb{i}}}} & {0.3602 + {0.4406{\mathbb{i}}}} & {0.3247 + {0.0797{\mathbb{i}}}} \\ {0.03536 - {0.3536{\mathbb{i}}}} & {0.3794 - {0.0965{\mathbb{i}}}} & {0.7586 + {0.0754{\mathbb{i}}}} & {0.3794 + {0.2846{\mathbb{i}}}} \\ {0 - {0.5000{\mathbb{i}}}} & {0.5609 - {0.3720{\mathbb{i}}}} & {0.2289 - {0.0434{\mathbb{i}}}} & {0.4262 - {0.5068{\mathbb{i}}}} \\ {{- 0.3536} - {0.3536{\mathbb{i}}}} & {{- 0.0965} - {0.3794{\mathbb{i}}}} & {0.0754 - {0.1870{\mathbb{i}}}} & {0.2846 - {0.3794{\mathbb{i}}}} \end{matrix}$

3. Third Scheme to Design a 6-Bit Codebook:

(1) Operation 1:

The following 4-bit codebook corresponding to the transmission rank 1 may be obtained from the 4-bit codebook disclosed in the above Table 1. In this example, the 4-bit codebook corresponding to the transmission rank 1, disclosed in the above Table 1, is referred to as a base codebook: base_cbk(:,1:16)=[C _(1,1) . . . C _(16,1)] with C _(i,1) taken from table 1

(2) Operation 2:

Eight vectors included in two DFT matrices may be added to the base codebook of operation 1, so that the base codebook may include 24 vectors. An exemplary process for adding the vectors to the base codebook is as follows:

Nmat=4;% number of DFT matrices     indexmat = repmat(0:3,4,1);     DFT_mat = exp(j*2*pi*indexmat.*(indexmat′)/4)/sqrt(4);     for counter = 1:Nmat       offsnow = exp(j*2*pi*(0:(3))*(counter−1)/(4*Nmat));       Rnow = diag(offsnow);   Wbis(1:4,1:4,counter) = Rnow*DFT_mat;  end;  % rank 1  % 8 more precoders  base_cbk(:,17:20)= Wbis(:,[1:4],2); base_cbk(:,21:24)= Wbis(:,[1:4],4);

As shown in ‘base_cbk(:,17:20)=Wbis(:,[1:4],2);’ and ‘base_cbk(:,21:24)=Wbis(:,[1:4],4);’, 24 vectors may be included in the base codebook.

A local codebook local_cbk may be defined, for example, as follows:

${{local\_ cbk} = \begin{bmatrix} \begin{matrix} {0.0975 +} \\ {0.4904{\mathbb{i}}} \end{matrix} & \begin{matrix} {0.0975 +} \\ {0.4904{\mathbb{i}}} \end{matrix} & \begin{matrix} {0.0975 +} \\ {0.4904{\mathbb{i}}} \end{matrix} & \begin{matrix} {0.0975 +} \\ {0.4904{\mathbb{i}}} \end{matrix} \\ \begin{matrix} {{- 0.4904} +} \\ {0.0975{\mathbb{i}}} \end{matrix} & \begin{matrix} {{- 0.0975} -} \\ {0.4904{\mathbb{i}}} \end{matrix} & \begin{matrix} {0.4904 -} \\ {0.0975{\mathbb{i}}} \end{matrix} & \begin{matrix} {0.0975 +} \\ {0.4904{\mathbb{i}}} \end{matrix} \\ \begin{matrix} {{- 0.4904} +} \\ {0.0975{\mathbb{i}}} \end{matrix} & \begin{matrix} {0.4904 -} \\ {0.0975{\mathbb{i}}} \end{matrix} & \begin{matrix} {{- 0.4904} +} \\ {0.0975{\mathbb{i}}} \end{matrix} & \begin{matrix} {0.4904 -} \\ {0.0975{\mathbb{i}}} \end{matrix} \\ \begin{matrix} {0.3536 +} \\ {0.3536{\mathbb{i}}} \end{matrix} & \begin{matrix} {0.3536 -} \\ {0.3536{\mathbb{i}}} \end{matrix} & \begin{matrix} {{- 0.3536} -} \\ {{- 0.3536}{\mathbb{i}}} \end{matrix} & \begin{matrix} {{- 0.3536} +} \\ {0.3536{\mathbb{i}}} \end{matrix} \end{bmatrix}};$

(3) Operation 3:

The local codebook local_cbk may be scaled to obtain the localized codebook localized_cbk. An exemplary scaling process is as follows:

r=abs(local_cbk); phase_local_cbk=local_cbk./r; alpha=0.9835;

Here, the localized codebook localized_cbk may be expressed, for example, as follows:

localized_cbk= [sqrt(1−alpha{circumflex over ( )}2*(1−(r(1,1)){circumflex over ( )}2))*phase_local_cbk(1,1),sqrt(1− alpha{circumflex over ( )}2*(1−(r(1,2)){circumflex over ( )}2))*phase_local_cbk(1,2),sqrt(1− alpha{circumflex over ( )}2*(1−(r(1,3)){circumflex over ( )}2))*phase_local_cbk(1,3),sqrt(1− alpha{circumflex over ( )}2*(1−(r(1,4)){circumflex over ( )}2))*phase_local_cbk(1,4);... alpha*r(2,1)*phase_local_cbk(2,1),alpha*r(2,2)* phase_local_cbk(2,2),alpha*r(2,3)*phase_local_cb k(2,3),alpha*r(2,4)*phase_local_cbk(2,4);... alpha*r(3,1)*phase_local_cbk(3,1),alpha*r(3,2)* phase_local_cbk(3,2),alpha*r(3,3)*phase_local_cb k(3,3),alpha*r(3,4)*phase_local_cbk(3,4);... alpha*r(4,1)*phase_local_cbk(4,1),alpha*r(4,2)* phase_local_cbk(4,2),alpha*r(4,3)*phase_local_cb k(4,3),alpha*r(4,4)*phase_local_cbk(4,4)];

Where the localized codebook localized_cbk is calculated through the aforementioned process, vectors of the localized codebook localized_cbk may be normalized, for example, through the following exemplary process:

for k=1:size(localized_cbk,2)     localized_cbk(:,k)=localized_cbk(:,k)/     norm(localized_cbk(:,k));   end

(4) Operation 4:

A final codebook final_cbk may be obtained by rotating the normalized localized codebook localized_cbk around the base codebook, for example, through the following exemplary process:

for k=1:24       [U,S,V]=svd(base_cbk(:,k));       R=U′*V; % U′*V*base_cbk(:,1,k)=[1;0;0;0]       if k<=16         rotated_localized=R'*localized_cbk(:,1:2);%, where only first two vectors of localized_cbk are rotated and the base codebook is maintained as a centroid.    final_cbk_rank{1}(:,1,(k−1)*3+1:k*3)=    [base_cbk(:,k),rotated_localized];       else         rotated_localized=R'*localized_cbk(:,1);%, where only first two vectors of localized_cbk are rotated, and the base codebook is maintained as a centroid.         final_cbk_rank{1}(:,1,16*3+(k−17)*2+1:16*3+(k− 16)*2)=[base_cbk(:,k),rotated_localized];       end     end;

6 bits of final rank 1 codebook may be given, for example, by final_cbk_rank{1}.

A final rank 2 codebook, a final rank 3 codebook, and/or a final rank 4 codebook may also be obtained based on the final rank 1 codebook. For example, unitary matrices including columns of the final rank 1 codebook, different from the first 16 vectors of the base codebook, may be obtained. Thus, a total of 48 unitary matrices may be obtained. A total of 54 matrices may be obtained if W₁ through W₆, are added to the 48 unitary matrices. The 54 matrices may become elements of the final rank 4 codebook.

The final rank 2 codebook may be obtained by taking the first two columns from the 48 unitary matrices, and taking 16 matrices or codewords from the 4-bit rank 2 codebook disclosed in the above Table 1.

The final rank 3 codebook may be obtained by taking the second through the fourth columns from the 48 unitary matrices, and taking 16 matrices or codewords from the 4-bit rank 3 codebook disclosed in the above Table 1.

6 bits of the final rank 1 codebook, the final rank 2 codebook, the final rank 3 codebook, and the final rank 4 codebook may be expressed, for example, as follows:

-   -   Final Rank 1 Codebook:

${V\; 1\left( {\text{:},\text{:},1} \right)} = \begin{matrix} 0.5000 \\ 0.5000 \\ 0.5000 \\ 0.5000 \end{matrix}$ ${V\; 1\left( {\text{:},\text{:},2} \right)} = \begin{matrix} {{- 0.2573} + {0.5267{\mathbb{i}}}} \\ {0.4306 + {0.2510{\mathbb{i}}}} \\ {0.4306 + {0.2510{\mathbb{i}}}} \\ {{- 0.3995} - {0.0009{\mathbb{i}}}} \end{matrix}$ ${V\; 1\left( {\text{:},\text{:},3} \right)} = \begin{matrix} {0.4182 - {0.2060{\mathbb{i}}}} \\ {0.2693 + {0.5849{\mathbb{i}}}} \\ {{- 0.3088} + {0.1985{\mathbb{i}}}} \\ {{- 0.1743} + {0.4504{\mathbb{i}}}} \end{matrix}$ ${V\; 1\left( {\text{:},\text{:},4} \right)} = \begin{matrix} 0.5000 \\ {- 0.5000} \\ 0.5000 \\ {- 0.5000} \end{matrix}$

${V\; 1\left( {\text{:},\text{:},5} \right)} = \begin{matrix} {{- 0.1228} + {0.0831{\mathbb{i}}}} \\ {0.4892 - {0.2949{\mathbb{i}}}} \\ {0.4754 + {0.1031{\mathbb{i}}}} \\ {{- 0.3409} - {0.5467{\mathbb{i}}}} \end{matrix}$ ${V\; 1\left( {\text{:},\text{:},6} \right)} = \begin{matrix} {0.1663 + {0.6240{\mathbb{i}}}} \\ {0.0064 + {0.1030{\mathbb{i}}}} \\ {{- 0.3928} + {0.4752{\mathbb{i}}}} \\ {{- 0.4372} - {0.0316{\mathbb{i}}}} \end{matrix}$ ${V\; 1\left( {\text{:},\text{:},7} \right)} = \begin{matrix} {- 0.5000} \\ {- 0.5000} \\ 0.5000 \\ 0.5000 \end{matrix}$ ${V\; 1\left( {\text{:},\text{:},8} \right)} = \begin{matrix} {0.1228 - {0.0831{\mathbb{i}}}} \\ {{- 0.4754} - {0.1031{\mathbb{i}}}} \\ {{- 0.4892} + {0.2949{\mathbb{i}}}} \\ {0.3409 + {0.5467{\mathbb{i}}}} \end{matrix}$

${V\; 1\left( {\text{:},\text{:},9} \right)} = \begin{matrix} {0.4119 - {0.2376{\mathbb{i}}}} \\ {0.0073 - {0.7328{\mathbb{i}}}} \\ {0.3791 + {0.1546{\mathbb{i}}}} \\ {0.2445 - {0.0972{\mathbb{i}}}} \end{matrix}$ ${V\; 1\left( {\text{:},\text{:},10} \right)} = \begin{matrix} {- 0.5000} \\ 0.5000 \\ 0.5000 \\ {- 0.5000} \end{matrix}$ ${V\; 1\left( {\text{:},\text{:},11} \right)} = \begin{matrix} {{- 0.7073} - {0.3349{\mathbb{i}}}} \\ {{- 0.2125} + {0.3788{\mathbb{i}}}} \\ {{- 0.2125} + {0.3788{\mathbb{i}}}} \\ {0.0779 + {0.0648{\mathbb{i}}}} \end{matrix}$ ${V\; 1\left( {\text{:},\text{:},12} \right)} = \begin{matrix} {{- 0.0318} - {0.3722{\mathbb{i}}}} \\ {{- 0.0512} - {0.1869{\mathbb{i}}}} \\ {0.5269 + {0.1995{\mathbb{i}}}} \\ {0.3031 - {0.6431{\mathbb{i}}}} \end{matrix}$

${V\; 1\left( {\text{:},\text{:},13} \right)} = \begin{matrix} 0.5000 \\ {0 + {0.5000{\mathbb{i}}}} \\ 0.5000 \\ {0 + {0.5000{\mathbb{i}}}} \end{matrix}$ ${V\; 1\left( {\text{:},\text{:},14} \right)} = \begin{matrix} {0.0318 + {0.3722{\mathbb{i}}}} \\ {0.1869 - {0.0512{\mathbb{i}}}} \\ {0.5269 + {0.1995{\mathbb{i}}}} \\ {{- 0.6431} - {0.3031{\mathbb{i}}}} \end{matrix}$ ${V\; 1\left( {\text{:},\text{:},15} \right)} = \begin{matrix} {{- 0.1228} + {0.0831{\mathbb{i}}}} \\ {{- 0.1031} + {0.4754{\mathbb{i}}}} \\ {{- 0.4892} + {0.2949{\mathbb{i}}}} \\ {{- 0.5467} + {0.3409{\mathbb{i}}}} \end{matrix}$ ${V\; 1\left( {\text{:},\text{:},16} \right)} = \begin{matrix} 0.5000 \\ {0 - {0.5000{\mathbb{i}}}} \\ 0.5000 \\ {0 - {0.5000{\mathbb{i}}}} \end{matrix}$

${V\; 1\left( {\text{:},\text{:},17} \right)} = \begin{matrix} {{- 0.4119} + {0.2376{\mathbb{i}}}} \\ {0.7328 + {0.0073{\mathbb{i}}}} \\ {0.3791 + {0.1546{\mathbb{i}}}} \\ {{- 0.0972} - {0.2445{\mathbb{i}}}} \end{matrix}$ ${V\; 1\left( {\text{:},\text{:},18} \right)} = \begin{matrix} {0.7073 + {0.3349{\mathbb{i}}}} \\ {0.3788 + {0.2125{\mathbb{i}}}} \\ {{- 0.2125} + {0.3788{\mathbb{i}}}} \\ {{- 0.0648} + {0.0779{\mathbb{i}}}} \end{matrix}$ ${V\; 1\left( {\text{:},\text{:},19} \right)} = {{\begin{matrix} {- 0.5000} \\ {0 - {0.5000{\mathbb{i}}}} \\ 0.5000 \\ {0 + {0.5000{\mathbb{i}}}} \end{matrix}V\; 1\left( {\text{:},\text{:},20} \right)} = \begin{matrix} {{- 0.1663} - {0.6240{\mathbb{i}}}} \\ {{- 0.1030} + {0.0064{\mathbb{i}}}} \\ {{- 0.3928} + {0.4752{\mathbb{i}}}} \\ {{- 0.0316} + {0.4372{\mathbb{i}}}} \end{matrix}}$

${V\; 1\left( {\text{:},\text{:},21} \right)} = \begin{matrix} {0.2473 - {0.5267{\mathbb{i}}}} \\ {0.2510 - {04306{\mathbb{i}}}} \\ {0.4306 + {0.2510{\mathbb{i}}}} \\ {0.0009 - {0.3995{\mathbb{i}}}} \end{matrix}$ ${V\; 1\left( {\text{:},\text{:},22} \right)} = \begin{matrix} {- 0.5000} \\ {0 + {0.5000{\mathbb{i}}}} \\ 0.5000 \\ {0 - {0.5000{\mathbb{i}}}} \end{matrix}$ ${V\; 1\left( {\text{:},\text{:},23} \right)} = \begin{matrix} {{- 0.4182} + {0.2060{\mathbb{i}}}} \\ {{- 0.5849} + {0.2693{\mathbb{i}}}} \\ {{- 0.3088} + {0.1985{\mathbb{i}}}} \\ {0.4504 + {0.1743{\mathbb{i}}}} \end{matrix}$ ${V\; 1\left( {\text{:},\text{:},24} \right)} = \begin{matrix} {0.1228 - {0.0831{\mathbb{i}}}} \\ {{- 0.2949} - {0.4892{\mathbb{i}}}} \\ {0.4754 + {0.1031{\mathbb{i}}}} \\ {0.5467 - {0.3409{\mathbb{i}}}} \end{matrix}$

${V\; 1\left( {\text{:},\text{:},25} \right)} = \begin{matrix} \begin{matrix} \begin{matrix} 0.5000 \\ 0.5000 \end{matrix} \\ 0.5000 \end{matrix} \\ {- 0.5000} \end{matrix}$ ${V\; 1\left( {\text{:},\text{:},26} \right)} = \begin{matrix} {{- 0.6051} + {0.1790{\mathbb{i}}}} \\ {0.3147 + {0.1351{\mathbb{i}}}} \\ {0.3147 + {0.1351{\mathbb{i}}}} \\ {{- 0.1801} - {0.5787{\mathbb{i}}}} \end{matrix}$ ${V\; 1\left( {\text{:},\text{:},27} \right)} = \begin{matrix} \begin{matrix} \begin{matrix} {0.0704 + {0.1417{\mathbb{i}}}} \\ {0.1534 + {0.7008{\mathbb{i}}}} \end{matrix} \\ {{- 0.4248} + {0.3144{\mathbb{i}}}} \end{matrix} \\ {{- 0.4053} + {0.1292{\mathbb{i}}}} \end{matrix}$ ${V\; 1\left( {\text{:},\text{:},28} \right)} = \begin{matrix} \begin{matrix} \begin{matrix} 0.5000 \\ {0 + {0.5000{\mathbb{i}}}} \end{matrix} \\ {- 0.5000} \end{matrix} \\ {0 + {0.5000{\mathbb{i}}}} \end{matrix}$

${V\; 1\left( {\text{:},\text{:},29} \right)} = \begin{matrix} \begin{matrix} \begin{matrix} {0.5141 + {0.2763{\mathbb{i}}}} \\ {0.2189 + {0.1095{\mathbb{i}}}} \end{matrix} \\ {0.2769 - {0.3593{\mathbb{i}}}} \end{matrix} \\ {{- 0.6111} - {0.1423{\mathbb{i}}}} \end{matrix}$ ${V\; 1\left( {\text{:},\text{:},30} \right)} = \begin{matrix} \begin{matrix} \begin{matrix} {{- 0.6051} + {0.1790{\mathbb{i}}}} \\ {{- 0.1351} + {0.3147{\mathbb{i}}}} \end{matrix} \\ {{- 0.3147} - {0.1351{\mathbb{i}}}} \end{matrix} \\ {{- 0.5787} + {0.1801{\mathbb{i}}}} \end{matrix}$ ${V\; 1\left( {\text{:},\text{:},31} \right)} = \begin{matrix} \begin{matrix} \begin{matrix} 0.5000 \\ {- 0.5000} \end{matrix} \\ 0.5000 \end{matrix} \\ 0.5000 \end{matrix}$ ${V\; 1\left( {\text{:},\text{:},32} \right)} = \begin{matrix} \begin{matrix} \begin{matrix} {0.2250 + {0.4308{\mathbb{i}}}} \\ {0.3732 - {0.4108{\mathbb{i}}}} \end{matrix} \\ {0.5913 + {0.2190{\mathbb{i}}}} \end{matrix} \\ {{- 0.2387} - {0.0328{\mathbb{i}}}} \end{matrix}$

${V\; 1\left( {\text{:},\text{:},33} \right)} = \begin{matrix} \begin{matrix} \begin{matrix} {0.5141 + {0.2763{\mathbb{i}}}} \\ {{- 0.1095} + {0.2189{\mathbb{i}}}} \end{matrix} \\ {{- 0.2769} + {0.3593{\mathbb{i}}}} \end{matrix} \\ {{- 0.1423} + {0.6111{\mathbb{i}}}} \end{matrix}$ ${V\; 1\left( {\text{:},\text{:},34} \right)} = \begin{matrix} \begin{matrix} \begin{matrix} 0.5000 \\ {0 - {0.5000{\mathbb{i}}}} \end{matrix} \\ {- 0.5000} \end{matrix} \\ {0 - {0.5000{\mathbb{i}}}} \end{matrix}$ ${V\; 1\left( {\text{:},\text{:},35} \right)} = \begin{matrix} \begin{matrix} \begin{matrix} {0.0704 + {0.1417{\mathbb{i}}}} \\ {0.7008 - {0.1534{\mathbb{i}}}} \end{matrix} \\ {0.4248 - {0.3144{\mathbb{i}}}} \end{matrix} \\ {{- 0.1292} - {0.4053{\mathbb{i}}}} \end{matrix}$ ${V\; 1\left( {\text{:},\text{:},36} \right)} = \begin{matrix} \begin{matrix} \begin{matrix} {0.2250 + {0.4308{\mathbb{i}}}} \\ {0.4108 + {0.3732{\mathbb{i}}}} \end{matrix} \\ {{- 0.5913} - {0.2190{\mathbb{i}}}} \end{matrix} \\ {{- 0.0328} + {0.2387{\mathbb{i}}}} \end{matrix}$

${V\; 1\left( {\text{:},\text{:},37} \right)} = \begin{matrix} \begin{matrix} \begin{matrix} 0.5000 \\ {0.3536 + {0.3536{\mathbb{i}}}} \end{matrix} \\ {0 + {0.5000{\mathbb{i}}}} \end{matrix} \\ {{- 0.3536} + {03536{\mathbb{i}}}} \end{matrix}$ ${V\; 1\left( {\text{:},\text{:},38} \right)} = \begin{matrix} \begin{matrix} \begin{matrix} {{- 0.0376} + {0.4566{\mathbb{i}}}} \\ {0.2688 + {0.1481{\mathbb{i}}}} \end{matrix} \\ {0.1588 - {0.0744{\mathbb{i}}}} \end{matrix} \\ {{- 0.5917} - {0.5613{\mathbb{i}}}} \end{matrix}$ ${V\; 1\left( {\text{:},\text{:},39} \right)} = \begin{matrix} \begin{matrix} \begin{matrix} {{- 0.4472} - {0.1208{\mathbb{i}}}} \\ {{- 0.0781} + {0.4936{\mathbb{i}}}} \end{matrix} \\ {{- 0.6133} - {0.0191{\mathbb{i}}}} \end{matrix} \\ {{- 0.3591} + {0.1738{\mathbb{i}}}} \end{matrix}$ ${V\; 1\left( {\text{:},\text{:},40} \right)} = \begin{matrix} \begin{matrix} \begin{matrix} 0.5000 \\ {{- 0.3536} + {0.3536{\mathbb{i}}}} \end{matrix} \\ {0 - {0.5000{\mathbb{i}}}} \end{matrix} \\ {0.3536 + {0.3536{\mathbb{i}}}} \end{matrix}$

${V\; 1\left( {\text{:},\text{:},41} \right)} = \begin{matrix} \begin{matrix} \begin{matrix} {0.4535 + {0.1524{\mathbb{i}}}} \\ {0.1943 - {0.1220{\mathbb{i}}}} \end{matrix} \\ {0.7044 - {0.2811{\mathbb{i}}}} \end{matrix} \\ {{- 0.3738} - {0.0597{\mathbb{i}}}} \end{matrix}$ ${V\; 1\left( {\text{:},\text{:},42} \right)} = \begin{matrix} \begin{matrix} \begin{matrix} {{- 0.0376} + {0.4566{\mathbb{i}}}} \\ {{- 0.1481} + {0.2688{\mathbb{i}}}} \end{matrix} \\ {{- 0.1588} + {0.0744{\mathbb{i}}}} \end{matrix} \\ {{- 0.5613} + {0.5917{\mathbb{i}}}} \end{matrix}$ ${V\; 1\left( {\text{:},\text{:},43} \right)} = \begin{matrix} \begin{matrix} \begin{matrix} 0.5000 \\ {{- 0.3536} - {0.3536{\mathbb{i}}}} \end{matrix} \\ {0 + {0.5000{\mathbb{i}}}} \end{matrix} \\ {0.3536 - {0.3536{\mathbb{i}}}} \end{matrix}$ ${V\; 1\left( {\text{:},\text{:},44} \right)} = \begin{matrix} \begin{matrix} \begin{matrix} {0.2356 + {0.5396{\mathbb{i}}}} \\ {0.6510 - {0.4238{\mathbb{i}}}} \end{matrix} \\ {0.1311 + {0.0167{\mathbb{i}}}} \end{matrix} \\ {{- 0.0198} - {0.1791{\mathbb{i}}}} \end{matrix}$

${V\; 1\left( {\text{:},\text{:},45} \right)} = \begin{matrix} \begin{matrix} \begin{matrix} {0.4535 + {0.1524{\mathbb{i}}}} \\ {0.1220 + {0.1943{\mathbb{i}}}} \end{matrix} \\ {{- 0.7044} + {0.2811{\mathbb{i}}}} \end{matrix} \\ {{- 0.0597} + {0.3738{\mathbb{i}}}} \end{matrix}$ ${V\; 1\left( {\text{:},\text{:},46} \right)} = \begin{matrix} \begin{matrix} \begin{matrix} 0.5000 \\ {0.3536 - {0.3536{\mathbb{i}}}} \end{matrix} \\ {0 - {0.5000{\mathbb{i}}}} \end{matrix} \\ {{- 0.3536} - {0.3536{\mathbb{i}}}} \end{matrix}$ ${V\; 1\left( {\text{:},\text{:},47} \right)} = \begin{matrix} \begin{matrix} \begin{matrix} {{- 0.4472} - {0.1208{\mathbb{i}}}} \\ {0.4936 + {0.0781{\mathbb{i}}}} \end{matrix} \\ {0.6133 + {0.0191{\mathbb{i}}}} \end{matrix} \\ {{- 0.1738} - {0.3591{\mathbb{i}}}} \end{matrix}$ ${V\; 1\left( {\text{:},\text{:},48} \right)} = \begin{matrix} \begin{matrix} \begin{matrix} {0.2356 + {0.5396{\mathbb{i}}}} \\ {0.4238 + {0.6510{\mathbb{i}}}} \end{matrix} \\ {{- 0.1311} - {0.0167{\mathbb{i}}}} \end{matrix} \\ {{- 0.1791} + {0.0198{\mathbb{i}}}} \end{matrix}$

${V\; 1\left( {\text{:},\text{:},49} \right)} = \begin{matrix} \begin{matrix} \begin{matrix} 0.5000 \\ {0.4619 + {0.1913{\mathbb{i}}}} \end{matrix} \\ {0.3536 + {0.3536{\mathbb{i}}}} \end{matrix} \\ {0.1913 + {0.4619{\mathbb{i}}}} \end{matrix}$ ${V\; 1\left( {\text{:},\text{:},50} \right)} = \begin{matrix} \begin{matrix} \begin{matrix} {{- 0.0628} + {0.5038{\mathbb{i}}}} \\ {0.3646 + {0.2226{\mathbb{i}}}} \end{matrix} \\ {0.2517 + {0.1533{\mathbb{i}}}} \end{matrix} \\ {{- 0.6562} - {0.2058{\mathbb{i}}}} \end{matrix}$ ${V\; 1\left( {\text{:},\text{:},51} \right)} = \begin{matrix} \begin{matrix} \begin{matrix} 0.5000 \\ {{- 0.1913} + {0.4619{\mathbb{i}}}} \end{matrix} \\ {{- 0.3536} - {0.3536{\mathbb{i}}}} \end{matrix} \\ {0.4619 - {0.1913{\mathbb{i}}}} \end{matrix}$ ${V\; 1\left( {\text{:},\text{:},52} \right)} = \begin{matrix} \begin{matrix} \begin{matrix} {0.4184 + {0.4842{\mathbb{i}}}} \\ {0.1085 - {0.0629{\mathbb{i}}}} \end{matrix} \\ {0.5948 - {0.4538{\mathbb{i}}}} \end{matrix} \\ {{- 0.0601} - {0.1068{\mathbb{i}}}} \end{matrix}$

${V\; 1\left( {\text{:},\text{:},53} \right)} = \begin{matrix} \begin{matrix} \begin{matrix} 0.5000 \\ {{- 0.4619} - {0.1913{\mathbb{i}}}} \end{matrix} \\ {0.3536 + {0.3536{\mathbb{i}}}} \end{matrix} \\ {{- 0.1913} - {0.4619{\mathbb{i}}}} \end{matrix}$ ${V\; 1\left( {\text{:},\text{:},54} \right)} = \begin{matrix} \begin{matrix} \begin{matrix} {{- 0.1083} + {0.4189{\mathbb{i}}}} \\ {0.6032 - {0.3824{\mathbb{i}}}} \end{matrix} \\ {0.2610 + {0.1225{\mathbb{i}}}} \end{matrix} \\ {{- 0.0597} - {0.4649{\mathbb{i}}}} \end{matrix}$ ${V\; 1\left( {\text{:},\text{:},55} \right)} = \begin{matrix} \begin{matrix} \begin{matrix} 0.5000 \\ {0.1913 - {0.4619{\mathbb{i}}}} \end{matrix} \\ {{- 0.3536} - {0.3536{\mathbb{i}}}} \end{matrix} \\ {{- 0.4619} + {0.1913{\mathbb{i}}}} \end{matrix}$ ${V\; 1\left( {\text{:},\text{:},56} \right)} = \begin{matrix} \begin{matrix} \begin{matrix} {{- 0.0430} - {0.3791{\mathbb{i}}}} \\ {0.5314 - {0.0969{\mathbb{i}}}} \end{matrix} \\ {0.5001 - {0.1416{\mathbb{i}}}} \end{matrix} \\ {{- 0.3832} - {0.3817{\mathbb{i}}}} \end{matrix}$

${V\; 1\left( {\text{:},\text{:},57} \right)} = \begin{matrix} \begin{matrix} \begin{matrix} 0.5000 \\ {0.1913 + {0.4619`{\mathbb{i}}}} \end{matrix} \\ {{- 0.3536} + {0.3536{\mathbb{i}}}} \end{matrix} \\ {{- 0.4619} - {0.1913{\mathbb{i}}}} \end{matrix}$ ${V\; 1\left( {\text{:},\text{:},58} \right)} = \begin{matrix} \begin{matrix} \begin{matrix} {{- 0.0197} + {0.5406{\mathbb{i}}}} \\ {0.1681 + {0.0640{\mathbb{i}}}} \end{matrix} \\ {0.2143 - {0.3250{\mathbb{i}}}} \end{matrix} \\ {{- 0.2387} - {0.6830{\mathbb{i}}}} \end{matrix}$ ${V\; 1\left( {\text{:},\text{:},59} \right)} = \begin{matrix} \begin{matrix} \begin{matrix} 0.5000 \\ {{- 0.4619} + {0.1913{\mathbb{i}}}} \end{matrix} \\ {0.3536 - {0.3536{\mathbb{i}}}} \end{matrix} \\ {{- 0.1913} + {0.4619{\mathbb{i}}}} \end{matrix}$ ${V\; 1\left( {\text{:},\text{:},60} \right)} = \begin{matrix} \begin{matrix} \begin{matrix} {0.1819 - {0.0589{\mathbb{i}}}} \\ {0.3368 - {0.1998{\mathbb{i}}}} \end{matrix} \\ {0.6565 - {0.0556{\mathbb{i}}}} \end{matrix} \\ {{- 0.5241} - {0.3183{\mathbb{i}}}} \end{matrix}$

${V\; 1\left( {\text{:},\text{:},61} \right)} = \begin{matrix} \begin{matrix} \begin{matrix} 0.5000 \\ {{- 0.1913} - {0.4619{\mathbb{i}}}} \end{matrix} \\ {{- 0.3536} + {0.3536{\mathbb{i}}}} \end{matrix} \\ {0.4619 + {0.1913{\mathbb{i}}}} \end{matrix}$ ${V\; 1\left( {\text{:},\text{:},62} \right)} = \begin{matrix} \begin{matrix} \begin{matrix} {0.5307 + {0.2465{\mathbb{i}}}} \\ {0.6357 - {0.3878{\mathbb{i}}}} \end{matrix} \\ {0.1539 - {0.1260{\mathbb{i}}}} \end{matrix} \\ {{- 0.2499} - {0.0328{\mathbb{i}}}} \end{matrix}$ ${V\; 1\left( {\text{:},\text{:},63} \right)} = \begin{matrix} \begin{matrix} \begin{matrix} 0.5000 \\ {0.4619 - {0.1913{\mathbb{i}}}} \end{matrix} \\ {0.3536 - {0.3536{\mathbb{i}}}} \end{matrix} \\ {0.1913 - {0.4619{\mathbb{i}}}} \end{matrix}$ ${V\; 1\left( {\text{:},\text{:},64} \right)} = \begin{matrix} \begin{matrix} \begin{matrix} {{- 0.4886} + {0.2995{\mathbb{i}}}} \\ {0.4671 + {0.2039{\mathbb{i}}}} \end{matrix} \\ {0.5829 + {0.1869{\mathbb{i}}}} \end{matrix} \\ {{- 0.1465} - {0.1250{\mathbb{i}}}} \end{matrix}$

-   -   Final Rank 2 Codebook:

${V\; 2\left( {\text{:},\text{:},1} \right)} = \begin{matrix} 0.5000 & 0.5000 \\ 0.5000 & {- 0.5000} \\ 0.5000 & 0.5000 \\ 0.5000 & {- 0.5000} \end{matrix}$ ${V\; 2\left( {\text{:},\text{:},2} \right)} = {\begin{matrix} \begin{matrix} \begin{matrix} {{- 0.2573} + {0.5267{\mathbb{i}}}} \\ {0.4306 + {0.2510{\mathbb{i}}}} \end{matrix} \\ {0.4306 + {0.2510{\mathbb{i}}}} \end{matrix} \\ {{- 0.3995} - {0.0009{\mathbb{i}}}} \end{matrix}\begin{matrix} \begin{matrix} \begin{matrix} {{- 0.4516} - {0.0164{\mathbb{i}}}} \\ {{- 0.3631} - {0.4030{\mathbb{i}}}} \end{matrix} \\ {0.4237 + {0.2733{\mathbb{i}}}} \end{matrix} \\ {0.2522 + {0.4286{\mathbb{i}}}} \end{matrix}}$ ${V\; 2\left( {\text{:},\text{:},3} \right)} = \begin{matrix} {0.4182 - {0.2060{\mathbb{i}}}} & {0.7071 - {0.3537{\mathbb{i}}}} \\ {0.2693 + {0.5849{\mathbb{i}}}} & {{- 0.1549} - {0.4387{\mathbb{i}}}} \\ {{- 0.3088} + {0.1985{\mathbb{i}}}} & {0.3107 - {0.1994{\mathbb{i}}}} \\ {{- 0.1743} + {0.4504{\mathbb{i}}}} & {{- 0.1065} + {0.1039{\mathbb{i}}}} \end{matrix}$ ${V\; 2\left( {\text{:},\text{:},4} \right)} = \begin{matrix} 0.5000 & {- 0.5000} \\ 0.5000 & {- 0.5000} \\ 0.5000 & 0.5000 \\ 0.5000 & 0.5000 \end{matrix}$

${V\; 2\left( {\text{:},\text{:},5} \right)} = \begin{matrix} {{- 0.1228} + {0.0831{\mathbb{i}}}} & {0.1911 + {0.1844{\mathbb{i}}}} \\ {0.4892 - {0.2949{\mathbb{i}}}} & {{- 0.6076} - {0.2220{\mathbb{i}}}} \\ {0.4754 + {0.1031{\mathbb{i}}}} & {0.0935 - {0.1294{\mathbb{i}}}} \\ {{- 0.3409} - {0.5467{\mathbb{i}}}} & {0.3520 - {0.6014{\mathbb{i}}}} \end{matrix}$ ${V\; 2\left( {\text{:},\text{:},6} \right)} = \begin{matrix} {0.1663 + {0.6240{\mathbb{i}}}} & {{- 0.4390} - {0.0177{\mathbb{i}}}} \\ {0.0064 + {0.1030{\mathbb{i}}}} & {{- 0.2573} - {0.1623{\mathbb{i}}}} \\ {{- 0.3928} + {0.4752{\mathbb{i}}}} & {0.4550 + {0.6115{\mathbb{i}}}} \\ {{- 0.4372} - {0.0316{\mathbb{i}}}} & {0.0477 - {0.3623{\mathbb{i}}}} \end{matrix}$ ${V\; 2\left( {\text{:},\text{:},7} \right)} = \begin{matrix} 0.5000 & {- 0.5000} \\ 0.5000 & 0.5000 \\ 0.5000 & 0.5000 \\ 0.5000 & {- 0.5000} \end{matrix}$ ${V\; 2\left( {\text{:},\text{:},8} \right)} = \begin{matrix} {0.1228 - {0.0831{\mathbb{i}}}} & {{- 0.8432} - {0.1719{\mathbb{i}}}} \\ {{- 0.4754} - {0.1031{\mathbb{i}}}} & {0.0004 - {0.0643{\mathbb{i}}}} \\ {{- 0.4892} + {0.2949{\mathbb{i}}}} & {0.0542 + {0.3715{\mathbb{i}}}} \\ {0.3409 + {0.5467{\mathbb{i}}}} & {{- 0.2612} + {0.2147{\mathbb{i}}}} \end{matrix}$

${V\; 2\left( {\text{:},\text{:},9} \right)} = \begin{matrix} {0.4119 - {0.2376{\mathbb{i}}}} & {0.2307 - {0.3523{\mathbb{i}}}} \\ {0.0073 - {0.7328{\mathbb{i}}}} & {{- 0.1995} - {0.0992{\mathbb{i}}}} \\ {0.3791 + {0.1546{\mathbb{i}}}} & {{- 0.6261} + {0.4891{\mathbb{i}}}} \\ {0.2445 - {0.0972{\mathbb{i}}}} & {{- 0.3749} - {0.0349{\mathbb{i}}}} \end{matrix}$ ${V\; 2\left( {\text{:},\text{:},10} \right)} = \begin{matrix} 0.5000 & {- 0.5000} \\ {- 0.5000} & {- 0.5000} \\ 0.5000 & 0.5000 \\ {- 0.5000} & 0.5000 \end{matrix}$ ${V\; 2\left( {\text{:},\text{:},11} \right)} = \begin{matrix} {{- 0.7073} - {0.3349{\mathbb{i}}}} & {{- 0.3380} - {0.1860{\mathbb{i}}}} \\ {{- 0.2125} + {0.3788{\mathbb{i}}}} & {{- 0.2536} - {0.2825{\mathbb{i}}}} \\ {{- 0.2125} + {0.3788{\mathbb{i}}}} & {0.6905 - {0.2237{\mathbb{i}}}} \\ {0.0779 + {0.0648{\mathbb{i}}}} & {{- 0.3771} + {0.1950{\mathbb{i}}}} \end{matrix}$ ${V\; 2\left( {\text{:},\text{:},12} \right)} = \begin{matrix} {{- 0.0318} - {0.3722{\mathbb{i}}}} & {0.5870 + {0.5837{\mathbb{i}}}} \\ {{- 0.0512} - {0.1869{\mathbb{i}}}} & {0.1353 - {0.0302{\mathbb{i}}}} \\ {0.5269 + {0.1995{\mathbb{i}}}} & {0.1672 - {0.4033{\mathbb{i}}}} \\ {0.3031 - {0.6431{\mathbb{i}}}} & {0.1619 - {0.2805`{\mathbb{i}}}} \end{matrix}$

${V\; 2\left( {\text{:},\text{:},13} \right)} = \begin{matrix} 0.5000 & {- 0.5000} \\ {- 0.5000} & 0.5000 \\ 0.5000 & 0.5000 \\ {- 0.5000} & {- 0.5000} \end{matrix}$ ${V\; 2\left( {\text{:},\text{:},14} \right)} = \begin{matrix} {0.0318 + {0.3722{\mathbb{i}}}} & {0.2910 + {0.0846{\mathbb{i}}}} \\ {0.1869 - {0.0512{\mathbb{i}}}} & {{- 0.5838} - {0.6826{\mathbb{i}}}} \\ {0.5269 + {0.1995{\mathbb{i}}}} & {{- 0.1084} + {0.1034{\mathbb{i}}}} \\ {{- 0.6431} - {0.3031{\mathbb{i}}}} & {0.0232 - {0.2799{\mathbb{i}}}} \end{matrix}$ ${V\; 2\left( {\text{:},\text{:},15} \right)} = \begin{matrix} {{- 0.1228} + {0.0831{\mathbb{i}}}} & {0.4864 + {0.1062{\mathbb{i}}}} \\ {{- 0.1031} + {0.4754{\mathbb{i}}}} & {{- 0.3579} + {0.0709{\mathbb{i}}}} \\ {{- 0.4892} + {0.2949{\mathbb{i}}}} & {0.3743 - {0.4217{\mathbb{i}}}} \\ {{- 0.5467} + {0.3409{\mathbb{i}}}} & {{- 0.2104} + {0.5067{\mathbb{i}}}} \end{matrix}$ ${V\; 2\left( {\text{:},\text{:},16} \right)} = \begin{matrix} {- 0.5000} & {- 0.5000} \\ {- 0.5000} & 0.5000 \\ 0.5000 & 0.5000 \\ 0.5000 & {- 0.5000} \end{matrix}$

${V\; 2\left( {\text{:},\text{:},17} \right)} = \begin{matrix} {{- 0.4119} + {0.2376{\mathbb{i}}}} & {{- 0.3475} + {0.0349{\mathbb{i}}}} \\ {0.7328 + {0.0073{\mathbb{i}}}} & {{- 0.3183} + {0.2743{\mathbb{i}}}} \\ {0.3791 + {0.1546{\mathbb{i}}}} & {0.3910 - {0.0682{\mathbb{i}}}} \\ {{- 0.0972} - {0.2445{\mathbb{i}}}} & {{- 0.5729} + {0.4644{\mathbb{i}}}} \end{matrix}$ ${V\; 2\left( {\text{:},\text{:},18} \right)} = \begin{matrix} {0.7073 + {0.3349{\mathbb{i}}}} & {0.3580 + {0.4230{\mathbb{i}}}} \\ {0.3788 + {0.2125{\mathbb{i}}}} & {{- 0.5016} - {0.3133{\mathbb{i}}}} \\ {{- 0.2125} + {0.3788{\mathbb{i}}}} & {0.4516 - {0.0120{\mathbb{i}}}} \\ {{- 0.0648} + {0.0779{\mathbb{i}}}} & {0.2277 - {0.2953{\mathbb{i}}}} \end{matrix}$ ${V\; 2\left( {\text{:},\text{:},19} \right)} = \begin{matrix} 0.5000 & {- 0.5000} \\ {0 + {0.5000{\mathbb{i}}}} & {0 - {0.5000{\mathbb{i}}}} \\ 0.5000 & 0.5000 \\ {0 + {0.5000{\mathbb{i}}}} & {0 + {0.5000{\mathbb{i}}}} \end{matrix}$ ${V\; 2\left( {\text{:},\text{:},20} \right)} = \begin{matrix} {{- 0.1663} - {0.6240{\mathbb{i}}}} & {{- 0.0285} - {0.3000{\mathbb{i}}}} \\ {{- 0.1030} + {0.0064{\mathbb{i}}}} & {{- 0.0146} + {0.4747{\mathbb{i}}}} \\ {{- 0.3928} + {0.4752{\mathbb{i}}}} & {0.1862 + {0.2754{\mathbb{i}}}} \\ {{- 0.0316} + {0.4372{\mathbb{i}}}} & {{- 0.4436} - {0.6134{\mathbb{i}}}} \end{matrix}$

${V\; 2\left( {\text{:},\text{:},21} \right)} = \begin{matrix} {0.2573 - {0.5267{\mathbb{i}}}} & {0.4288 + {0.0624{\mathbb{i}}}} \\ {0.2510 - {0.4306{\mathbb{i}}}} & {{- 0.2155} - {0.5900{\mathbb{i}}}} \\ {0.4306 + {0.2510{\mathbb{i}}}} & {0.0071 - {0.1770{\mathbb{i}}}} \\ {0.0009 - {0.3995{\mathbb{i}}}} & {0.1913 + {0.5913{\mathbb{i}}}} \end{matrix}$ ${V\; 2\left( {\text{:},\text{:},22} \right)} = \begin{matrix} 0.5000 & {- 0.5000} \\ {0 + {0.5000{\mathbb{i}}}} & {0 + {0.5000{\mathbb{i}}}} \\ 0.5000 & 0.5000 \\ {0 + {0.5000{\mathbb{i}}}} & {0 - {0.5000{\mathbb{i}}}} \end{matrix}$ ${V\; 2\left( {\text{:},\text{:},23} \right)} = \begin{matrix} {{- 0.4182} + {0.2060{\mathbb{i}}}} & {{- 0.0368} - {0.04444{\mathbb{i}}}} \\ {{- 0.5849} + {0.2693{\mathbb{i}}}} & {0.0931 + {0.1023{\mathbb{i}}}} \\ {{- 0.3088} + {0.1985{\mathbb{i}}}} & {0.6755 + {0.2582{\mathbb{i}}}} \\ {0.4504 + {0.1743{\mathbb{i}}}} & {0.1401 + {0.6595{\mathbb{i}}}} \end{matrix}$ ${V\; 2\left( {\text{:},\text{:},24} \right)} = \begin{matrix} {0.1228 - {0.0831{\mathbb{i}}}} & {0.5267 + {0.1302{\mathbb{i}}}} \\ {{- 0.2949} - {0.4892{\mathbb{i}}}} & {{- 0.3082} + {0.1349{\mathbb{i}}}} \\ {0.4754 + {0.1031{\mathbb{i}}}} & {{- 0.5798} - {0.2378{\mathbb{i}}}} \\ {0.5467 - {0.3409{\mathbb{i}}}} & {0.4428 + {0.0607{\mathbb{i}}}} \end{matrix}$

${V\; 2\left( {\text{:},\text{:},25} \right)} = \begin{matrix} 0.5000 & {- 0.5000} \\ {0 - {0.5000{\mathbb{i}}}} & {0 - {0.5000{\mathbb{i}}}} \\ 0.5000 & 0.5000 \\ {0 - {0.5000{\mathbb{i}}}} & {0 + {0.5000{\mathbb{i}}}} \end{matrix}$ ${V\; 2\left( {\text{:},\text{:},26} \right)} = \begin{matrix} {{- 0.6051} + {0.1790{\mathbb{i}}}} & {{- 0.2867} + {0.4568{\mathbb{i}}}} \\ {0.3147 + {0.1351{\mathbb{i}}}} & {{- 0.3377} + {0.4028{\mathbb{i}}}} \\ {0.3147 + {0.1351{\mathbb{i}}}} & {{- 0.4367} - {0.4408{\mathbb{i}}}} \\ {{- 0.1801} - {0.5787{\mathbb{i}}}} & {0.2118 - {0.0548{\mathbb{i}}}} \end{matrix}$ ${V\; 2\left( {\text{:},\text{:},27} \right)} = \begin{matrix} {0.0704 + {0.1417{\mathbb{i}}}} & {0.4489 - {0.4294{\mathbb{i}}}} \\ {0.1534 + {0.7008{\mathbb{i}}}} & {0.0676 - {0.2514{\mathbb{i}}}} \\ {{- 0.4248} + {0.3144{\mathbb{i}}}} & {{- 0.0562} - {0.3389{\mathbb{i}}}} \\ {{- 0.4053} + {0.1292{\mathbb{i}}}} & {{- 0.6352} + {0.1576{\mathbb{i}}}} \end{matrix}$ ${V\; 2\left( {\text{:},\text{:},28} \right)} = \begin{matrix} 0.5000 & {- 0.5000} \\ {0 - {0.5000{\mathbb{i}}}} & {0 + {0.5000{\mathbb{i}}}} \\ 0.5000 & 0.5000 \\ {0 - {0.5000{\mathbb{i}}}} & {0 - {0.5000{\mathbb{i}}}} \end{matrix}$

${V\; 2\left( {\text{:},\text{:},29} \right)} = \begin{matrix} {0.5141 + {0.2763{\mathbb{i}}}} & {0.0224 - {0.0857{\mathbb{i}}}} \\ {0.2189 + {0.1095{\mathbb{i}}}} & {0.4370 + {0.0749{\mathbb{i}}}} \\ {0.2769 - {0.3593{\mathbb{i}}}} & {0.0927 - {0.6134{\mathbb{i}}}} \\ {{- 0.6111} - {0.1423{\mathbb{i}}}} & {0.6031 - {0.2167{\mathbb{i}}}} \end{matrix}$ ${V\; 2\left( {\text{:},\text{:},30} \right)} = \begin{matrix} {{- 0.6051} + {0.1790{\mathbb{i}}}} & {{- 0.2021} + {0.2515{\mathbb{i}}}} \\ {{- 0.1351} + {0.3147{\mathbb{i}}}} & {{- 0.2703} - {0.8413{\mathbb{i}}}} \\ {{- 0.3147} - {0.1351{\mathbb{i}}}} & {{- 0.3054} - {0.0715{\mathbb{i}}}} \\ {{- 0.5787} + {0.1801{\mathbb{i}}}} & {0.1019 + {0.0787{\mathbb{i}}}} \end{matrix}$ ${V\; 2\left( {\text{:},\text{:},31} \right)} = \begin{matrix} 0.5000 & {- 0.5000} \\ 0.5000 & {0 - {0.5000{\mathbb{i}}}} \\ 0.5000 & 0.5000 \\ 0.5000 & {0 + {0.5000{\mathbb{i}}}} \end{matrix}$ ${V\; 2\left( {\text{:},\text{:},32} \right)} = \begin{matrix} {0.2250 + {0.4308{\mathbb{i}}}} & {{- 0.0056} + {0.0131{\mathbb{i}}}} \\ {0.3732 - {0.4108{\mathbb{i}}}} & {0.0211 + {0.5772{\mathbb{i}}}} \\ {0.5913 + {0.2190{\mathbb{i}}}} & {0.6155 - {0.2919{\mathbb{i}}}} \\ {{- 0.2387} - {0.0328{\mathbb{i}}}} & {0.3534 - {0.2780{\mathbb{i}}}} \end{matrix}$

${V\; 2\left( {\text{:},\text{:},33} \right)} = \begin{matrix} {0.5141 + {0.2763{\mathbb{i}}}} & {0.0913 - {0.3481{\mathbb{i}}}} \\ {{- 0.1095} + {0.2189{\mathbb{i}}}} & {{- 0.5047} + {0.4712{\mathbb{i}}}} \\ {{- 0.2769} + {0.3593{\mathbb{i}}}} & {0.0947 - {0.5855{\mathbb{i}}}} \\ {{- 0.1423} + {0.6111{\mathbb{i}}}} & {{- 0.0727} + {0.1915{\mathbb{i}}}} \end{matrix}$ ${V\; 2\left( {\text{:},\text{:},34} \right)} = \begin{matrix} 0.5000 & {- 0.5000} \\ 0.5000 & {0 + {0.5000{\mathbb{i}}}} \\ 0.5000 & 0.5000 \\ 0.5000 & {0 - {0.5000{\mathbb{i}}}} \end{matrix}$ ${V\; 2\left( {\text{:},\text{:},35} \right)} = \begin{matrix} {0.0704 + {0.1417{\mathbb{i}}}} & {0.2602 - {0.3335{\mathbb{i}}}} \\ {0.7008 - {0.1534{\mathbb{i}}}} & {0.3267 - {0.3041{\mathbb{i}}}} \\ {0.4248 - {0.3144{\mathbb{i}}}} & {0.1781 + {0.6675{\mathbb{i}}}} \\ {{- 0.1292} - {0.4053{\mathbb{i}}}} & {{- 0.1799} + {0.3348{\mathbb{i}}}} \end{matrix}$ ${V\; 2\left( {\text{:},\text{:},36} \right)} = \begin{matrix} {0.2250 + {0.4308{\mathbb{i}}}} & {{- 0.1449} - {0.3470{\mathbb{i}}}} \\ {0.4108 + {0.3732{\mathbb{i}}}} & {{- 0.0692} + {0.7493{\mathbb{i}}}} \\ {{- 0.5913} - {0.2190{\mathbb{i}}}} & {{- 0.1504} + {0.4516{\mathbb{i}}}} \\ {{- 0.0328} + {0.2387{\mathbb{i}}}} & {0.1073 - {0.2330{\mathbb{i}}}} \end{matrix}$

${V\; 2\left( {\text{:},\text{:},37} \right)} = \begin{matrix} 0.5000 & {- 0.5000} \\ {0 + {0.5000{\mathbb{i}}}} & {- 0.5000} \\ 0.5000 & 0.5000 \\ {0 + {0.5000{\mathbb{i}}}} & 0.5000 \end{matrix}$ ${V\; 2\left( {\text{:},\text{:},38} \right)} = \begin{matrix} {{- 0.0376} + {0.4566{\mathbb{i}}}} & {{- 0.6780} - {0.3635{\mathbb{i}}}} \\ {0.2688 + {0.1481{\mathbb{i}}}} & {0.1160 + {0.0644{\mathbb{i}}}} \\ {0.1588 - {0.0744{\mathbb{i}}}} & {{- 0.3301} + {0.2541{\mathbb{i}}}} \\ {{- 0.5917} - {0.5613{\mathbb{i}}}} & {{- 0.4384} + {0.1574{\mathbb{i}}}} \end{matrix}$ ${V\; 2\left( {\text{:},\text{:},39} \right)} = \begin{matrix} {{- 0.4472} - {0.1208{\mathbb{i}}}} & {0.2383 + {0.0963{\mathbb{i}}}} \\ {{- 0.0781} + {0.4936{\mathbb{i}}}} & {0.2810 + {0.0700{\mathbb{i}}}} \\ {{- 0.6133} - {0.0191{\mathbb{i}}}} & {0.2857 - {0.4528{\mathbb{i}}}} \\ {{- 0.3591} + {0.1738{\mathbb{i}}}} & {{- 0.4779} + {0.5788{\mathbb{i}}}} \end{matrix}$ ${V\; 2\left( {\text{:},\text{:},40} \right)} = \begin{matrix} 0.5000 & {- 0.5000} \\ {0 + {0.5000{\mathbb{i}}}} & 0.5000 \\ 0.5000 & 0.5000 \\ {0 + {0.5000{\mathbb{i}}}} & {- 0.5000} \end{matrix}$

${V\; 2\left( {\text{:},\text{:},41} \right)} = \begin{matrix} {0.4535 + {0.1524{\mathbb{i}}}} & {0.6361 - {0.0379{\mathbb{i}}}} \\ {0.1943 - {0.1220{\mathbb{i}}}} & {{- 0.3164} + {0.0577{\mathbb{i}}}} \\ {0.7044 - {0.2811{\mathbb{i}}}} & {0.0562 + {0.0352{\mathbb{i}}}} \\ {{- 0.3738} - {0.0597{\mathbb{i}}}} & {0.6782 - {0.1616{\mathbb{i}}}} \end{matrix}$ ${V\; 2\left( {\text{:},\text{:},42} \right)} = \begin{matrix} {{- 0.0376} + {0.4566{\mathbb{i}}}} & {{- 0.6004} - {0.0993{\mathbb{i}}}} \\ {{- 0.1481} + {0.2688{\mathbb{i}}}} & {{- 0.4515} - {0.2985{\mathbb{i}}}} \\ {{- 0.1588} + {0.0744{\mathbb{i}}}} & {{- 0.1174} + {0.2648{\mathbb{i}}}} \\ {{- 0.5613} + {0.5917{\mathbb{i}}}} & {0.3667 + {0.3441{\mathbb{i}}}} \end{matrix}$ ${V\; 2\left( {\text{:},\text{:},43} \right)} = \begin{matrix} 0.5000 & 0.5000 \\ 0.5000 & {- 0.5000} \\ 0.5000 & 0.5000 \\ {- 0.5000} & 0.5000 \end{matrix}$ ${V\; 2\left( {\text{:},\text{:},44} \right)} = \begin{matrix} {0.2356 + {0.5396{\mathbb{i}}}} & {{- 0.2530} + {0.3209{\mathbb{i}}}} \\ {0.6510 - {0.4238{\mathbb{i}}}} & {{- 0.2885} - {0.0485{\mathbb{i}}}} \\ {0.1311 + {0.0167{\mathbb{i}}}} & {{- 0.0415} - {0.7379{\mathbb{i}}}} \\ {{- 0.0198} - {0.1791{\mathbb{i}}}} & {0.1649 - {0.4171{\mathbb{i}}}} \end{matrix}$

${V\; 2\left( {\text{:},\text{:},45} \right)} = \begin{matrix} {0.4535 + {0.1524{\mathbb{i}}}} & {0.2821 - {0.1279{\mathbb{i}}}} \\ {0.1220 + {0.1943{\mathbb{i}}}} & {{- 0.0004} - {0.0348{\mathbb{i}}}} \\ {{- 0.7044} + {0.2811{\mathbb{i}}}} & {{- 0.2923} + {0.0831{\mathbb{i}}}} \\ {{- 0.0597} + {0.3738{\mathbb{i}}}} & {{- 0.0749} - {0.8971{\mathbb{i}}}} \end{matrix}$ ${V\; 2\left( {\text{:},\text{:},46} \right)} = \begin{matrix} 0.5000 & 0.5000 \\ {{- 0.3536} + {0.3536{\mathbb{i}}}} & {0.3536 - {0.3536{\mathbb{i}}}} \\ {0 - {0.5000{\mathbb{i}}}} & {0 - {0.5000{\mathbb{i}}}} \\ {0.3536 + {0.3536{\mathbb{i}}}} & {{- 0.3536} - {0.3536{\mathbb{i}}}} \end{matrix}$ ${V\; 2\left( {\text{:},\text{:},47} \right)} = \begin{matrix} {{- 0.4472} - {0.1208{\mathbb{i}}}} & {{- 0.4748} + {0.1205{\mathbb{i}}}} \\ {0.4936 + {0.0781{\mathbb{i}}}} & {0.2258 + {0.2637{\mathbb{i}}}} \\ {0.6133 + {0.0191{\mathbb{i}}}} & {{- 0.2650} + {0.3289{\mathbb{i}}}} \\ {{- 0.1738} - {0.3591{\mathbb{i}}}} & {{- 0.2798} + {0.6188{\mathbb{i}}}} \end{matrix}$ ${V\; 2\left( {\text{:},\text{:},48} \right)} = \begin{matrix} {0.2356 + {0.5396{\mathbb{i}}}} & {{- 0.0180} - {0.6214{\mathbb{i}}}} \\ {0.4238 + {0.6510{\mathbb{i}}}} & {{- 0.0262} + {0.4269{\mathbb{i}}}} \\ {{- 0.1311} - {0.0167{\mathbb{i}}}} & {0.1087 + {0.3811{\mathbb{i}}}} \\ {{- 0.1791} + {0.0198{\mathbb{i}}}} & {{- 0.5069} + {0.1291{\mathbb{i}}}} \end{matrix}$

${V\; 2\left( {\text{:},\text{:},49} \right)} = \begin{matrix} 0.5000 & {0.0906 - {0.2217{\mathbb{i}}}} \\ {0.4619 + {0.1913{\mathbb{i}}}} & {{- 0.3471} + {0.3239{\mathbb{i}}}} \\ {0.3536 + {0.3536{\mathbb{i}}}} & {{- 0.2568} + {0.4202{\mathbb{i}}}} \\ {0.1913 + {0.4619{\mathbb{i}}}} & {0.6329 - {0.2724{\mathbb{i}}}} \end{matrix}$ ${V\; 2\left( {\text{:},\text{:},50} \right)} = \begin{matrix} {{- 0.0628} + {0.5038{\mathbb{i}}}} & {{- 0.2542} - {0.0969{\mathbb{i}}}} \\ {0.3646 + {0.2226{\mathbb{i}}}} & {0.1965 + {0.0675{\mathbb{i}}}} \\ {0.2517 + {0.1533{\mathbb{i}}}} & {0.8534 + {0.0091{\mathbb{i}}}} \\ {{- 0.6562} - {0.2058{\mathbb{i}}}} & {0.3805 + {0.0988{\mathbb{i}}}} \end{matrix}$ ${V\; 2\left( {\text{:},\text{:},51} \right)} = \begin{matrix} 0.5000 & {{- 0.0310} + {0.3127{\mathbb{i}}}} \\ {{- 0.1913} + {0.4619{\mathbb{i}}}} & {{- 0.4944} - {0.1906{\mathbb{i}}}} \\ {{- 0.3536} - {0.3536{\mathbb{i}}}} & {0.0394 + {0.1700{\mathbb{i}}}} \\ {0.4619 - {0.1913{\mathbb{i}}}} & {{- 0.1337} - {0.7565{\mathbb{i}}}} \end{matrix}$ ${V\; 2\left( {\text{:},\text{:},52} \right)} = \begin{matrix} {0.4184 + {0.4842{\mathbb{i}}}} & {0.0769 + {0.4899{\mathbb{i}}}} \\ {0.1085 - {0.0629{\mathbb{i}}}} & {0.0623 - {0.0897{\mathbb{i}}}} \\ {0.5948 - {0.4538{\mathbb{i}}}} & {{- 0.4402} + {0.1739{\mathbb{i}}}} \\ {{- 0.0601} - {0.1068{\mathbb{i}}}} & {{- 0.7025} - {0.1568{\mathbb{i}}}} \end{matrix}$

${V\; 2\left( {\text{:},\text{:},53} \right)} = \begin{matrix} 0.5000 & {{- 0.1844} + {0.1638{\mathbb{i}}}} \\ {{- 0.4619} - {0.1913{\mathbb{i}}}} & {0.3389 - {0.3843{\mathbb{i}}}} \\ {0.3536 + {0.3536{\mathbb{i}}}} & {0.4409 - {0.5467{\mathbb{i}}}} \\ {{- 0.1913} - {0.4619{\mathbb{i}}}} & {{- 0.1168} - {0.4120{\mathbb{i}}}} \end{matrix}$ ${V\; 2\left( {\text{:},\text{:},54} \right)} = \begin{matrix} {{- 0.1083} + {0.4189{\mathbb{i}}}} & {{- 0.1461} + {0.1056{\mathbb{i}}}} \\ {0.6032 - {0.3824{\mathbb{i}}}} & {0.1171 - {0.2010{\mathbb{i}}}} \\ {0.2610 + {0.1225{\mathbb{i}}}} & {{- 0.9157} - {0.2359{\mathbb{i}}}} \\ {{- 0.0597} - {0.4649{\mathbb{i}}}} & {{- 0.0670} - {0.1213{\mathbb{i}}}} \end{matrix}$ ${V\; 2\left( {\text{:},\text{:},55} \right)} = \begin{matrix} 0.5000 & {0.5295 + {0.2225{\mathbb{i}}}} \\ {0.1913 - {0.4619{\mathbb{i}}}} & {0.0632 + {0.0938{\mathbb{i}}}} \\ {{- 0.3536} - {0.3536{\mathbb{i}}}} & {{- 0.3442} + {0.5791{\mathbb{i}}}} \\ {{- 0.4619} + {0.1913{\mathbb{i}}}} & {0.1494 - {0.4256{\mathbb{i}}}} \end{matrix}$ ${V\; 2\left( {\text{:},\text{:},56} \right)} = \begin{matrix} {{- 0.0430} - {0.3791{\mathbb{i}}}} & {{- 0.2534} + {0.0743{\mathbb{i}}}} \\ {0.5314 - {0.0969{\mathbb{i}}}} & {{- 0.0802} + {0.0767{\mathbb{i}}}} \\ {0.5001 - {0.1416{\mathbb{i}}}} & {{- 0.3092} + {0.6079{\mathbb{i}}}} \\ {{- 0.3832} - {0.3817{\mathbb{i}}}} & {{- 0.6565} - {0.1479{\mathbb{i}}}} \end{matrix}$

${V\; 2\left( {\text{:},\text{:},57} \right)} = \begin{matrix} 0.5000 & {{- 0.4933} + {0.2493{\mathbb{i}}}} \\ {0.1913 + {0.4619{\mathbb{i}}}} & {{- 0.0119} - {0.1919{\mathbb{i}}}} \\ {{- 0.3536} + {0.3536{\mathbb{i}}}} & {0.2792 + {0.2603{\mathbb{i}}}} \\ {{- 0.4619} - {0.1913{\mathbb{i}}}} & {{- 0.5616} - {0.4432{\mathbb{i}}}} \end{matrix}$ ${V\; 2\left( {\text{:},\text{:},58} \right)} = \begin{matrix} {{- 0.0197} + {0.5406{\mathbb{i}}}} & {0.5987 + {0.4276{\mathbb{i}}}} \\ {0.1681 + {0.0640{\mathbb{i}}}} & {{- 0.2253} + {0.0427{\mathbb{i}}}} \\ {0.2143 - {0.3250{\mathbb{i}}}} & {{- 0.1122} + {0.5394{\mathbb{i}}}} \\ {{- 0.2387} - {0.6830{\mathbb{i}}}} & {0.2949 - {0.1252{\mathbb{i}}}} \end{matrix}$ ${V\; 2\left( {\text{:},\text{:},59} \right)} = \begin{matrix} 0.5000 & {{- 0.1299} - {0.1774{\mathbb{i}}}} \\ {{- 0.4619} + {0.1913{\mathbb{i}}}} & {{- 0.7371} - {0.3712{\mathbb{i}}}} \\ {0.3536 - {0.3536{\mathbb{i}}}} & {{- 0.1244} - {0.1207{\mathbb{i}}}} \\ {{- 0.1913} + {0.4619{\mathbb{i}}}} & {0.4090 - {0.2705{\mathbb{i}}}} \end{matrix}$ ${V\; 2\left( {\text{:},\text{:},60} \right)} = \begin{matrix} {0.1819 - {0.0589{\mathbb{i}}}} & {{- 0.3263} - {0.7161{\mathbb{i}}}} \\ {0.3368 - {0.1998{\mathbb{i}}}} & {{- 0.0538} + {0.0491{\mathbb{i}}}} \\ {0.6565 - {0.0556{\mathbb{i}}}} & {0.2749 - {0.2169{\mathbb{i}}}} \\ {{- 0.5241} - {0.3183{\mathbb{i}}}} & {0.4347 - {0.2526{\mathbb{i}}}} \end{matrix}$

${V\; 2\left( {\text{:},\text{:},61} \right)} = \begin{matrix} 0.5000 & {{- 0.3550} + {0.4986{\mathbb{i}}}} \\ {{- 0.1913} - {0.4619{\mathbb{i}}}} & {{- 0.5979} - {0.4466{\mathbb{i}}}} \\ {{- 0.3536} + {0.3536{\mathbb{i}}}} & {0.2014 + {0.0225{\mathbb{i}}}} \\ {0.4619 + {0.1913{\mathbb{i}}}} & {{- 0.1636} - {0.0232{\mathbb{i}}}} \end{matrix}$ ${V\; 2\left( {\text{:},\text{:},62} \right)} = \begin{matrix} {0.5307 + {0.2465{\mathbb{i}}}} & {0.4315 + {0.2977{\mathbb{i}}}} \\ {0.6357 - {0.3878{\mathbb{i}}}} & {{- 0.3022} + {0.0611{\mathbb{i}}}} \\ {0.1539 - {0.1260{\mathbb{i}}}} & {{- 0.4238} - {0.3362{\mathbb{i}}}} \\ {{- 0.249} - {0.0328{\mathbb{i}}}} & {0.3188 - {0.4857{\mathbb{i}}}} \end{matrix}$ ${V\; 2\left( {\text{:},\text{:},63} \right)} = \begin{matrix} 0.5000 & {0.6135 - {0.5466{\mathbb{i}}}} \\ {0.4619 - {0.1913{\mathbb{i}}}} & {{- 0.3336} + {0.1681{\mathbb{i}}}} \\ {0.3536 - {0.3536{\mathbb{i}}}} & {0.0923 + {0.1646{\mathbb{i}}}} \\ {0.1913 - {0.4619{\mathbb{i}}}} & {0.2389 + {0.3044{\mathbb{i}}}} \end{matrix}$ ${V\; 2\left( {\text{:},\text{:},64} \right)} = \begin{matrix} {{- 0.4886} + {0.2995{\mathbb{i}}}} & {{- 0.0643} + {0.2882{\mathbb{i}}}} \\ {0.4671 + {0.2039{\mathbb{i}}}} & {0.1360 - {0.4277{\mathbb{i}}}} \\ {0.5829 + {0.1869{\mathbb{i}}}} & {{- 0.4682} + {0.3860{\mathbb{i}}}} \\ {{- 0.1465} - {0.1250{\mathbb{i}}}} & {{- 0.2986} - {0.5040{\mathbb{i}}}} \end{matrix}$

-   -   Final Rank 3 Codebook:

$\mspace{79mu}{{V\; 3\left( {\text{:},\text{:},1} \right)} = \begin{matrix} 0.5000 & 0.5000 & {- 0.5000} \\ 0.5000 & {- 0.5000} & {- 0.5000} \\ 0.5000 & 0.5000 & 0.5000 \\ 0.5000 & {- 0.5000} & 0.5000 \end{matrix}}$ ${V\; 3\left( {\text{:},\text{:},2} \right)} = \begin{matrix} {{- 0.4516} - {0.0164{\mathbb{i}}}} & {{- 0.1786} + {0.5269{\mathbb{i}}}} & {{- 0.2985} - {0.2313{\mathbb{i}}}} \\ {{- 0.3631} - {0.4030{\mathbb{i}}}} & {{- 0.3685} - {0.1544{\mathbb{i}}}} & {0.1966 + {0.5090{\mathbb{i}}}} \\ {0.4237 + {0.2733{\mathbb{i}}}} & {{- 0.5798} - {0.0933{\mathbb{i}}}} & {{- 0.1123} - {0.3741{\mathbb{i}}}} \\ {0.2522 + {0.4286{\mathbb{i}}}} & {{- 0.3685} + {0.2240{\mathbb{i}}}} & {0.0615 + {0.6351{\mathbb{i}}}} \end{matrix}$ ${V\; 3\left( {\text{:},\text{:},3} \right)} = \begin{matrix} {0.7071 - {0.3537{\mathbb{i}}}} & {{- 0.0971} + {0.0208{\mathbb{i}}}} & {{- 0.2518} - {0.2904{\mathbb{i}}}} \\ {{- 0.1549} - {0.4387{\mathbb{i}}}} & {{- 0.4646} - {0.3323{\mathbb{i}}}} & {0.1746 + {0.1101{\mathbb{i}}}} \\ {0.3107 - {0.1994{\mathbb{i}}}} & {{- 0.1616} + {0.7129{\mathbb{i}}}} & {0.4094 + {0.1641{\mathbb{i}}}} \\ {{- 0.1065} + {0.1039{\mathbb{i}}}} & {{- 0.1624} + {0.3212{\mathbb{i}}}} & {{- 0.7503} - {0.2284{\mathbb{i}}}} \end{matrix}$ $\mspace{20mu}{{V\; 3\left( {\text{:},\text{:},4} \right)} = \begin{matrix} 0.5000 & 0.5000 & {- 0.5000} \\ 0.5000 & {- 0.5000} & 0.5000 \\ 0.5000 & 0.5000 & 0.5000 \\ 0.5000 & {- 0.5000} & {- 0.5000} \end{matrix}}$

${V\; 3\left( {\text{:},\text{:},5} \right)} = \begin{matrix} {0.1911 + {0.1844{\mathbb{i}}}} & {{- 0.2742} - {0.7346{\mathbb{i}}}} & {0.4333 - {0.3239{\mathbb{i}}}} \\ {{- 0.6076} - {0.2220{\mathbb{i}}}} & {0.1147 - {0.2232{\mathbb{i}}}} & {{- 0.0716} - {0.4327{\mathbb{i}}}} \\ {0.0935 - {0.1294{\mathbb{i}}}} & {{- 0.2568} - {0.4393{\mathbb{i}}}} & {{- 0.3198} + {0.6137{\mathbb{i}}}} \\ {0.3520 - {0.6014{\mathbb{i}}}} & {0.1308 - {0.2148{\mathbb{i}}}} & {{- 0.1887} - {0.0218{\mathbb{i}}}} \end{matrix}$ ${V\; 3\left( {\text{:},\text{:},6} \right)} = \begin{matrix} {{- 0.4390} - {0.0177{\mathbb{i}}}} & {0.1890 + {0.1915{\mathbb{i}}}} & {0.5399 - {0.1614{\mathbb{i}}}} \\ {{- 0.2573} - {0.1623{\mathbb{i}}}} & {0.5097 - {0.7079{\mathbb{i}}}} & {{- 0.1643} + {0.3299{\mathbb{i}}}} \\ {0.4550 + {0.6115{\mathbb{i}}}} & {0.1352 - {0.1381{\mathbb{i}}}} & {{- 0.0396} - {0.0100{\mathbb{i}}}} \\ {0.0477 - {0.3623{\mathbb{i}}}} & {{- 0.0601} - {0.3545{\mathbb{i}}}} & {0.1275 - {0.7271{\mathbb{i}}}} \end{matrix}$ $\mspace{20mu}{{V\; 3\left( {\text{:},\text{:},7} \right)} = \begin{matrix} 0.5000 & {- 0.5000} & {- 0.5000} \\ 0.5000 & {- 0.5000} & 0.5000 \\ 0.5000 & 0.5000 & 0.5000 \\ 0.5000 & 0.5000 & {- 0.5000} \end{matrix}}$ ${V\; 3\left( {\text{:},\text{:},8} \right)} = \begin{matrix} {{- 0.8432} + {0.1719{\mathbb{i}}}} & {0.1018 + {0.2933{\mathbb{i}}}} & {0.3119 - {0.2093{\mathbb{i}}}} \\ {0.0004 - {0.0643{\mathbb{i}}}} & {0.6544 - {0.1493{\mathbb{i}}}} & {0.3891 + {0.3966{\mathbb{i}}}} \\ {0.0542 + {0.3715{\mathbb{i}}}} & {{- 0.5068} + {0.3662{\mathbb{i}}}} & {0.2327 + {0.2963{\mathbb{i}}}} \\ {{- 0.2612} + {0.2147{\mathbb{i}}}} & {0.1621 - {0.1894{\mathbb{i}}}} & {{- 0.3150} + {0.5560{\mathbb{i}}}} \end{matrix}$

${V\; 3\left( {\text{:},\text{:},9} \right)} = \begin{matrix} {0.2307 - {0.3523{\mathbb{i}}}} & {{- 0.3946} + {0.2222{\mathbb{i}}}} & {0.3873 + {0.4914{\mathbb{i}}}} \\ {{- 0.1995} - {0.0992{\mathbb{i}}}} & {0.1664 - {0.1501{\mathbb{i}}}} & {{- 0.5320} + {0.2829{\mathbb{i}}}} \\ {{- 0.6261} + {0.4891{\mathbb{i}}}} & {0.1146 + {0.2907{\mathbb{i}}}} & {0.0580 + {0.3166{\mathbb{i}}}} \\ {{- 0.3749} - {0.0349{\mathbb{i}}}} & {{- 0.2411} - {0.7674{\mathbb{i}}}} & {0.3196 - {0.1994{\mathbb{i}}}} \end{matrix}$ $\mspace{20mu}{{V\; 3\left( {\text{:},\text{:},10} \right)} = \begin{matrix} 0.5000 & {- 0.5000} & {- 0.5000} \\ {- 0.5000} & {- 0.5000} & 0.5000 \\ 0.5000 & 0.5000 & 0.5000 \\ {- 0.5000} & 0.5000 & {- 0.5000} \end{matrix}}$ ${V\; 3\left( {\text{:},\text{:},11} \right)} = \begin{matrix} {{- 0.3380} - {0.1860{\mathbb{i}}}} & {0.2936 - {0.0670{\mathbb{i}}}} & {0.1233 + {0.3645{\mathbb{i}}}} \\ {{- 0.2536} - {0.2825{\mathbb{i}}}} & {{- 0.4027} + {0.1913{\mathbb{i}}}} & {{- 0.6695} + {0.1424{\mathbb{i}}}} \\ {0.6905 - {0.2237{\mathbb{i}}}} & {0.4699 + {0.1884{\mathbb{i}}}} & {{- 0.0830} + {0.162{\mathbb{i}}}} \\ {{- 0.3771} + {0.1950{\mathbb{i}}}} & {0.6721 + {0.0505{\mathbb{i}}}} & {{- 0.4095} - {0.4330{\mathbb{i}}}} \end{matrix}$ ${V\; 3\left( {\text{:},\text{:},12} \right)} = \begin{matrix} {0.5870 + {0.5837{\mathbb{i}}}} & {0.1694 - {0.3686{\mathbb{i}}}} & {{- 0.1020} - {0.0171{\mathbb{i}}}} \\ {0.1353 - {0.0302{\mathbb{i}}}} & {0.2036 + {0.5782{\mathbb{i}}}} & {{- 0.5358} + {0.5295{\mathbb{i}}}} \\ {0.1672 - {0.4033{\mathbb{i}}}} & {0.4862 - {0.2843{\mathbb{i}}}} & {{- 0.3698} - {0.1949{\mathbb{i}}}} \\ {0.1619 - {0.2805{\mathbb{i}}}} & {0.0806 + {0.3688{\mathbb{i}}}} & {0.4237 - {0.2601{\mathbb{i}}}} \end{matrix}$

${\mspace{20mu}{{{V\; 3\left( {\text{:},\text{:},13} \right)} = \begin{matrix} 0.5000 & 0.5000 & {- 0.5000} \\ {0 + {0.5000{\mathbb{i}}}} & {0 - {0.5000{\mathbb{i}}}} & {0 - {0.5000{\mathbb{i}}}} \\ 0.5000 & 0.5000 & 0.5000 \\ {0 + {0.5000{\mathbb{i}}}} & {0 - {0.5000{\mathbb{i}}}} & {0 + {0.5000{\mathbb{i}}}} \end{matrix}}{{V\; 3\left( {\text{:},\text{:},14} \right)} = \begin{matrix} {0.2910 + {0.0846{\mathbb{i}}}} & {0.0464 + {0.0232{\mathbb{i}}}} & {{- 0.8617} + {0.1530{\mathbb{i}}}} \\ {{- 0.5838} - {0.6826{\mathbb{i}}}} & {0.1164 + {0.0920{\mathbb{i}}}} & {{- 0.3062} - {0.1997{\mathbb{i}}}} \\ {{- 0.1084} + {0.1034{\mathbb{i}}}} & {{- 06815} - {0.4049{\mathbb{i}}}} & {0.0000 - {0.01781{\mathbb{i}}}} \\ {0.0232 - {0.2799{\mathbb{i}}}} & {{- 0.4827} - {0.3376{\mathbb{i}}}} & {{- 0.1746} + {0.1956{\mathbb{i}}}} \end{matrix}}{{V\; 3\left( {\text{:},\text{:}, 15} \right)} =}}\quad}{\quad{{\begin{matrix} {0.4864 + {0.1062{\mathbb{i}}}} & {{- 0.5138} + {0.1606{\mathbb{i}}}} & {0.6091 + {0.2633{\mathbb{i}}}} \\ {{- 0.3579} + {0.0709{\mathbb{i}}}} & {0.2824 - {0.4495{\mathbb{i}}}} & {0.5834 - {0.0898{\mathbb{i}}}} \\ {0.3743 - {0.4217{\mathbb{i}}}} & {{- 0.1744} - {0.4396{\mathbb{i}}}} & {{- 0.3251} - {0.1626{\mathbb{i}}}} \\ {{- 0.2104} + {0.5067{\mathbb{i}}}} & {{- 0.1120} + {0.4384{\mathbb{i}}}} & {{- 0.2030} - {0.1946{\mathbb{i}}}} \end{matrix}\mspace{20mu} V\; 3\left( {\text{:},\text{:},16} \right)} = \begin{matrix} 0.5000 & 0.5000 & {- 0.5000} \\ {0 + {0.5000{\mathbb{i}}}} & {0 - {0.5000{\mathbb{i}}}} & {0 + {0.5000{\mathbb{i}}}} \\ 0.5000 & 0.5000 & 0.5000 \\ {0 + {0.5000{\mathbb{i}}}} & {0 - {0.5000{\mathbb{i}}}} & {0 - {0.5000{\mathbb{i}}}} \end{matrix}}}$

${V\; 3\left( {\text{:},\text{:},17} \right)} = \begin{matrix} {{- 0.3475} + {0.0349{\mathbb{i}}}} & {{- 0.5492} + {0.0805{\mathbb{i}}}} & {0.4831 - {0.3324{\mathbb{i}}}} \\ {{- 0.3183} + {0.2743{\mathbb{i}}}} & {0.0020 - {0.0107{\mathbb{i}}}} & {0.5026 + {0.1833{\mathbb{i}}}} \\ {0.3910 - {0.0682{\mathbb{i}}}} & {{- 0.6621} - {0.4101{\mathbb{i}}}} & {{- 0.2328} - {0.1188{\mathbb{i}}}} \\ {{- 0.5729} + {0.4644{\mathbb{i}}}} & {{- 0.2236} - {0.1878{\mathbb{i}}}} & {{- 0.5285} + {0.1492{\mathbb{i}}}} \end{matrix}$ ${V\; 3\left( {\text{:},\text{:},18} \right)} = \begin{matrix} {0.3580 + {0.4230{\mathbb{i}}}} & {0.2367 + {0.0128{\mathbb{i}}}} & {0.0222 - {0.1544{\mathbb{i}}}} \\ {{- 0.5016} - {0.3133{\mathbb{i}}}} & {0.2038 - {0.2663{\mathbb{i}}}} & {{- 0.4319} + {0.4031{\mathbb{i}}}} \\ {0.4516 - {0.0120{\mathbb{i}}}} & {{- 0.3421} - {0.6404{\mathbb{i}}}} & {{- 0.0186} + {0.2825{\mathbb{i}}}} \\ {0.2277 - {0.2953{\mathbb{i}}}} & {0.5298 + {0.1531{\mathbb{i}}}} & {0.5491 + {0.4950{\mathbb{i}}}} \end{matrix}$ $\mspace{79mu}{{V\; 3\left( {\text{:},\text{:},19} \right)} = \begin{matrix} 0.500 & {- 0.5000} & {- 0.5000} \\ {0 + {0.5000{\mathbb{i}}}} & {0 - {0.5000{\mathbb{i}}}} & {0 + {0.5000{\mathbb{i}}}} \\ 0.5000 & 0.5000 & 0.5000 \\ {0 + {0.5000{\mathbb{i}}}} & {0 + {0.5000{\mathbb{i}}}} & {0 - {0.5000{\mathbb{i}}}} \end{matrix}}$

${V\; 3\left( {\text{:},\text{:},20} \right)} = \begin{matrix} {{- 0.0285} - {0.3000{\mathbb{i}}}} & {0.2482 - {0.1368{\mathbb{i}}}} & {{- 0.3683} + {0.5255{\mathbb{i}}}} \\ {{- 0.0146} + {0.4747{\mathbb{i}}}} & {0.8444 + {0.1708{\mathbb{i}}}} & {0.1466 - {0.0063{\mathbb{i}}}} \\ {0.1862 + {0.2754{\mathbb{i}}}} & {{- 0.1088} + {0.1084{\mathbb{i}}}} & {{- 0.6599} + {0.2244{\mathbb{i}}}} \\ {{- 0.4436} - {0.6134{\mathbb{i}}}} & {0.3812 - {0.0925{\mathbb{i}}}} & {{- 0.1957} - {0.2063{\mathbb{i}}}} \end{matrix}$ ${V\; 3\left( {\text{:},\text{:},21} \right)} = \begin{matrix} {0.4288 + {0.0624{\mathbb{i}}}} & {{- 0.4668} + {0.1127{\mathbb{i}}}} & {{- 0.4182} + {0.2512{\mathbb{i}}}} \\ {{- 0.2155} - {0.5900{\mathbb{i}}}} & {0.4068 - {0.1439{\mathbb{i}}}} & {{- 0.2850} - {0.2995{\mathbb{i}}}} \\ {0.0071 - {0.1770{\mathbb{i}}}} & {0.2316 - {0.2993{\mathbb{i}}}} & {0.0461 + {0.7582{\mathbb{i}}}} \\ {0.1913 + {0.5913{\mathbb{i}}}} & {0.6629 + {0.0244{\mathbb{i}}}} & {0.0964 + {0.0693{\mathbb{i}}}} \end{matrix}$ $\mspace{76mu}{{V\; 3\left( {\text{:},\text{:},22} \right)} = \begin{matrix} 0.5000 & {- 0.5000} & {- 0.5000} \\ {0 - {0.5000{\mathbb{i}}}} & {0 - {0.5000{\mathbb{i}}}} & {0 + {0.5000{\mathbb{i}}}} \\ 0.5000 & 0.5000 & 0.5000 \\ {0 - {0.5000{\mathbb{i}}}} & {0 + {0.5000{\mathbb{i}}}} & {0 - {0.5000{\mathbb{i}}}} \end{matrix}}$ ${V\; 3\left( {\text{:},\text{:},23} \right)} = \begin{matrix} {{- 0.0368} - {0.0444{\mathbb{i}}}} & {{- 0.1903} + {0.2671{\mathbb{i}}}} & {0.4025 - {0.7140{\mathbb{i}}}} \\ {0.0931 + {0.1023{\mathbb{i}}}} & {0.3098 - {0.6795{\mathbb{i}}}} & {{- 0.0892} + {0.0221{\mathbb{i}}}} \\ {0.6755 + {0.2582{\mathbb{i}}}} & {{- 0.0591} + {0.4709{\mathbb{i}}}} & {{- 0.0849} + {0.3313{\mathbb{i}}}} \\ {0.1401 + {0.6595{\mathbb{i}}}} & {0.3082 - {0.1202{\mathbb{i}}}} & {0.4248 - {0.1494{\mathbb{i}}}} \end{matrix}$

${V\; 3\left( {\text{:},\text{:},24} \right)} = \begin{matrix} {0.5267 + {0.1302{\mathbb{i}}}} & {{- 0.3106} - {0.1437{\mathbb{i}}}} & {0.7407 - {0.1338{\mathbb{i}}}} \\ {{- 0.3082} + {0.1349{\mathbb{i}}}} & {{- 0.4197} - {0.5805{\mathbb{i}}}} & {{- 0.1306} - {0.1743{\mathbb{i}}}} \\ {{- 0.5798} - {0.2378{\mathbb{i}}}} & {{- 0.5074} + {0.1972{\mathbb{i}}}} & {0.2367 + {0.1351{\mathbb{i}}}} \\ {0.4428 + {0.0607{\mathbb{i}}}} & {{- 0.2678} + {0.0417{\mathbb{i}}}} & {{- 0.5235} + {0.1939{\mathbb{i}}}} \end{matrix}$ $\mspace{79mu}{{V\; 3\left( {\text{:},\text{:},25} \right)} = \begin{matrix} 0.5000 & 0.5000 & {- 0.5000} \\ 0.5000 & {- 0.5000} & {0 - {0.5000{\mathbb{i}}}} \\ 0.5000 & 0.5000 & 0.5000 \\ 0.5000 & {- 0.5000} & {0 + {0.5000{\mathbb{i}}}} \end{matrix}}$ ${V\; 3\left( {\text{:},\text{:},26} \right)} = \begin{matrix} {{- 0.2867} + {0.4568{\mathbb{i}}}} & {{- 0.0929} + {0.0656{\mathbb{i}}}} & {{- 0.1384} - {0.5281{\mathbb{i}}}} \\ {{- 0.3377} + {0.4028{\mathbb{i}}}} & {0.6771 - {0.3179{\mathbb{i}}}} & {0.2166 + {0.0016{\mathbb{i}}}} \\ {{- 0.4367} - {0.4408{\mathbb{i}}}} & {{- 0.1123} - {0.3181{\mathbb{i}}}} & {{- 0.5156} - {0.3436{\mathbb{i}}}} \\ {0.2118 - {0.0548{\mathbb{i}}}} & {0.5502 + {0.1048{\mathbb{i}}}} & {{- 0.5105} - {0.1021{\mathbb{i}}}} \end{matrix}$ ${V\; 3\left( {\text{:},\text{:},27} \right)} = \begin{matrix} {0.4489 - {0.4294{\mathbb{i}}}} & {0.4657 + {0.4116{\mathbb{i}}}} & {0.2255 + {0.3897{\mathbb{i}}}} \\ {0.0676 - {0.2514{\mathbb{i}}}} & {{- 0.5317} - {0.2639{\mathbb{i}}}} & {0.1962 + {0.1631{\mathbb{i}}}} \\ {{- 0.0562} - {0.3389{\mathbb{i}}}} & {{- 0.0047} + {0.3844{\mathbb{i}}}} & {{- 0.1563} - {0.6561{\mathbb{i}}}} \\ {{- 0.6352} + {0.1576{\mathbb{i}}}} & {{- 0.0225} + {0.3362{\mathbb{i}}}} & {0.3199 + {0.4182{\mathbb{i}}}} \end{matrix}$

$\mspace{79mu}{{V\; 3\left( {\text{:},\text{:},28} \right)} = \begin{matrix} 0.5000 & {- 0.5000} & {- 0.5000} \\ 0.5000 & {0 - {0.5000{\mathbb{i}}}} & {0 + {0.5000{\mathbb{i}}}} \\ 0.5000 & 0.5000 & 0.5000 \\ 0.5000 & {0 + {0.5000{\mathbb{i}}}} & {0 - {0.5000{\mathbb{i}}}} \end{matrix}}$ ${V\; 3\left( {\text{:},\text{:},29} \right)} = \begin{matrix} {0.0224 - {0.0857{\mathbb{i}}}} & {{- 0.3110} + {0.4440{\mathbb{i}}}} & {0.3131 - {0.5095{\mathbb{i}}}} \\ {0.4370 + {0.0749{\mathbb{i}}}} & {0.5479 + {0.3445{\mathbb{i}}}} & {0.3368 + {0.4595{\mathbb{i}}}} \\ {0.0927 - {0.6134{\mathbb{i}}}} & {{- 0.4211} + {0.0796{\mathbb{i}}}} & {{- 0.0820} + {0.4679{\mathbb{i}}}} \\ {0.6031 - {0.2167{\mathbb{i}}}} & {{- 0.1108} + {0.3020{\mathbb{i}}}} & {{- 0.0056} - {0.3033{\mathbb{i}}}} \end{matrix}$ ${V\; 3\left( {\text{:},\text{:},30} \right)} = \begin{matrix} {{- 0.2021} + {0.2515{\mathbb{i}}}} & {{- 0.1048} + {0.3295{\mathbb{i}}}} & {0.4277 + {0.4419{\mathbb{i}}}} \\ {{- 0.2703} - {0.8413{\mathbb{i}}}} & {0.2200 + {0.2221{\mathbb{i}}}} & {{- 0.0633} - {0.0093{\mathbb{i}}}} \\ {{- 0.3054} - {0.0715{\mathbb{i}}}} & {{- 0.4718} - {0.1404{\mathbb{i}}}} & {0.2258 - {0.7007{\mathbb{i}}}} \\ {0.1019 + {0.0787{\mathbb{i}}}} & {0.3731 - {0.6334{\mathbb{i}}}} & {{- 0.2723} - {0.0393{\mathbb{i}}}} \end{matrix}$ $\mspace{79mu}{{V\; 3\left( {\text{:},\text{:},31} \right)} = \begin{matrix} 0.5000 & 0.5000 & {- 0.5000} \\ {0 + {0.5000{\mathbb{i}}}} & {0 - {0.5000{\mathbb{i}}}} & {- 0.5000} \\ 0.5000 & 0.5000 & 0.5000 \\ {0 + {0.5000{\mathbb{i}}}} & {0 - {0.5000{\mathbb{i}}}} & 0.5000 \end{matrix}}$

${V\; 3\left( {\text{:},\text{:},32} \right)} = \begin{matrix} {{- 0.0056} + {0.0131{\mathbb{i}}}} & {0.0379 - {0.7861{\mathbb{i}}}} & {0.2371 + {0.2967{\mathbb{i}}}} \\ {0.0211 + {0.5772{\mathbb{i}}}} & {0.4481 - {0.2070{\mathbb{i}}}} & {0.0255 - {0.3378{\mathbb{i}}}} \\ {0.6155 - {0.2919{\mathbb{i}}}} & {0.1617 + {0.2064{\mathbb{i}}}} & {{- 0.2637} - {0.0051{\mathbb{i}}}} \\ {0.3534 - {0.2780{\mathbb{i}}}} & {0.2588 + {0.0366{\mathbb{i}}}} & {0.7632 - {0.2983{\mathbb{i}}}} \end{matrix}$ ${V\; 3\left( {\text{:},\text{:},33} \right)} = \begin{matrix} {0.0913 - {0.3481{\mathbb{i}}}} & {0.5632 + {0.2802{\mathbb{i}}}} & {{- 0.1951} - {0.3100{\mathbb{i}}}} \\ {{- 0.5047} + {0.4712{\mathbb{i}}}} & {{- 0.0240} - {0.0899{\mathbb{i}}}} & {{- 0.2776} - {0.6145{\mathbb{i}}}} \\ {0.0947 - {0.5855{\mathbb{i}}}} & {{- 0.3176} - {0.3672{\mathbb{i}}}} & {{- 0.4474} - {0.0811{\mathbb{i}}}} \\ {{- 0.0727} + {0.1915{\mathbb{i}}}} & {0.4510 - {0.3955{\mathbb{i}}}} & {0.2488 + {0.3775{\mathbb{i}}}} \end{matrix}$ $\mspace{79mu}{{V\; 3\left( {\text{:},\text{:},34} \right)} = \begin{matrix} 0.5000 & {- 0.5000} & {- 0.5000} \\ {0 + {0.5000{\mathbb{i}}}} & {- 0.5000} & 0.5000 \\ 0.5000 & 0.5000 & 0.5000 \\ {0 + {0.5000{\mathbb{i}}}} & 0.5000 & {- 0.5000} \end{matrix}}$ ${V\; 3\left( {\text{:},\text{:},35} \right)} = \begin{matrix} {0.2602 - {0.3335{\mathbb{i}}}} & {0.1575 - {0.6399{\mathbb{i}}}} & {0.5258 + {0.2920{\mathbb{i}}}} \\ {0.3267 - {0.3041{\mathbb{i}}}} & {{- 0.2930} + {0.2159{\mathbb{i}}}} & {{- 0.2319} + {0.3158{\mathbb{i}}}} \\ {0.1781 + {0.6675{\mathbb{i}}}} & {0.1266 - {0.0025{\mathbb{i}}}} & {0.4330 - {0.1995{\mathbb{i}}}} \\ {{- 0.1799} + {0.3348{\mathbb{i}}}} & {{- 0.3623} - {0.5347{\mathbb{i}}}} & {{- 0.3289} + {0.3863{\mathbb{i}}}} \end{matrix}$

${V\; 3\left( {\text{:},\text{:},36} \right)} = \begin{matrix} {{- 0.1449} - {0.3470{\mathbb{i}}}} & {{- 0.3069} - {0.2742{\mathbb{i}}}} & {{- 0.4090} + {0.5345{\mathbb{i}}}} \\ {{- 0.0692} + {0.7493{\mathbb{i}}}} & {0.0914 + {0.0688{\mathbb{i}}}} & {{- 0.3010} - {0.1486{\mathbb{i}}}} \\ {{- 0.1504} + {0.4516{\mathbb{i}}}} & {0.1108 - {0.0595{\mathbb{i}}}} & {{- 0.1821} + {0.5717{\mathbb{i}}}} \\ {0.1073 - {0.2330{\mathbb{i}}}} & {0.4107 + {0.7956{\mathbb{i}}}} & {{- 0.1571} + {0.2230{\mathbb{i}}}} \end{matrix}$ $\mspace{85mu}{{V\; 3\left( {\text{:},\text{:},37} \right)} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 \\ 0.5000 & {0 + {0.5000{\mathbb{i}}}} & {- 0.5000} \\ 0.5000 & {- 0.5000} & 0.5000 \\ {- 0.5000} & {0 + {0.5000{\mathbb{i}}}} & 0.5000 \end{matrix}}$ ${V\; 3\left( {\text{:},\text{:},38} \right)} = \begin{matrix} {{- 0.6780} - {0.3635{\mathbb{i}}}} & {0.0238 - {0.3004{\mathbb{i}}}} & {{- 0.2666} - {0.1906{\mathbb{i}}}} \\ {0.1160 + {0.0644{\mathbb{i}}}} & {0.7450 + {0.3155{\mathbb{i}}}} & {{- 0.4820} + {0.0363{\mathbb{i}}}} \\ {{- 0.3301} + {0.2541{\mathbb{i}}}} & {0.4517 - {0.0242{\mathbb{i}}}} & {0.6746 - {0.3688{\mathbb{i}}}} \\ {{- 0.4384} + {0.1574{\mathbb{i}}}} & {0.1584 + {0.1580{\mathbb{i}}}} & {{- 0.2294} + {0.1232{\mathbb{i}}}} \end{matrix}$ ${V\; 3\left( {\text{:},\text{:},39} \right)} = \begin{matrix} {0.2383 + {0.0963{\mathbb{i}}}} & {{- 0.0567} - {0.0816{\mathbb{i}}}} & {0.8144 + {0.2149{\mathbb{i}}}} \\ {0.2810 + {0.0700{\mathbb{i}}}} & {{- 0.5121} + {0.5153{\mathbb{i}}}} & {{- 0.1356} + {0.3467{\mathbb{i}}}} \\ {0.2857 - {0.4528{\mathbb{i}}}} & {0.4309 + {0.0872{\mathbb{i}}}} & {{- 0.3612} + {0.1144{\mathbb{i}}}} \\ {{- 0.4779} + {0.5788{\mathbb{i}}}} & {0.3586 + {0.3748{\mathbb{i}}}} & {{- 0.0135} - {0.0906{\mathbb{i}}}} \end{matrix}$

$\mspace{79mu}{{V\; 3\left( {\text{:},\text{:},40} \right)} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 \\ 0.5000 & {- 0.5000} & {0 - {0.5000{\mathbb{i}}}} \\ 0.5000 & 0.5000 & {- 0.5000} \\ {- 0.5000} & 0.5000 & {0 - {0.5000{\mathbb{i}}}} \end{matrix}}$ ${V\; 3\left( {\text{:},\text{:},41} \right)} = \begin{matrix} {0.6361 - {0.0379{\mathbb{i}}}} & {0.0303 + {0.1370{\mathbb{i}}}} & {{- 0.3832} + {0.4456{\mathbb{i}}}} \\ {{- 0.3164} + {0.0577{\mathbb{i}}}} & {0.8288 + {0.2462{\mathbb{i}}}} & {{- 0.2979} - {0.0876{\mathbb{i}}}} \\ {0.0562 + {0.0352{\mathbb{i}}}} & {{- 0.1507} - {0.2734{\mathbb{i}}}} & {0.0430 - {0.5667{\mathbb{i}}}} \\ {0.6782 - {0.1616{\mathbb{i}}}} & {0.3675 - {0.0175{\mathbb{i}}}} & {0.1704 - {0.4541{\mathbb{i}}}} \end{matrix}$ ${V\; 3\left( {\text{:},\text{:},42} \right)} = \begin{matrix} {{- 0.6004} - {0.0993{\mathbb{i}}}} & {0.1028 - {0.0939{\mathbb{i}}}} & {0.6203 + {0.1250{\mathbb{i}}}} \\ {{- 0.4515} - {0.2985{\mathbb{i}}}} & {{- 0.0297} + {0.3112{\mathbb{i}}}} & {{- 0.5456} - {0.4663{\mathbb{i}}}} \\ {{- 0.1174} + {0.2648{\mathbb{i}}}} & {{- 0.8756} + {0.2308{\mathbb{i}}}} & {{- 0.0070} + {0.2556{\mathbb{i}}}} \\ {0.3667 + {0.3441{\mathbb{i}}}} & {0.1195 - {0.2205{\mathbb{i}}}} & {{- 0.0920} - {0.1031{\mathbb{i}}}} \end{matrix}$ ${V\; 3\left( {\text{:},\text{:},43} \right)} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 \\ {0.3536 + {0.3536{\mathbb{i}}}} & {{- 0.3536} + {0.3536{\mathbb{i}}}} & {0.3536 - {0.3536{\mathbb{i}}}} \\ {0 + {0.5000{\mathbb{i}}}} & {0 - {0.5000{\mathbb{i}}}} & {0 - {0.5000{\mathbb{i}}}} \\ {{- 0.3536} + {0.3536{\mathbb{i}}}} & {0.3536 + {0.3536{\mathbb{i}}}} & {{- 0.3536} - {0.3536{\mathbb{i}}}} \end{matrix}$

${V\; 3\left( {\text{:},\text{:},44} \right)} = \begin{matrix} {{- 0.2530} + {0.3209{\mathbb{i}}}} & {{- 0.4814} + {0.3797{\mathbb{i}}}} & {{- 0.1906} - {0.2722{\mathbb{i}}}} \\ {{- 0.2885} - {0.0485{\mathbb{i}}}} & {{- 0.4053} - {0.3563{\mathbb{i}}}} & {0.1239 + {0.0673{\mathbb{i}}}} \\ {{- 0.0415} - {0.7379{\mathbb{i}}}} & {{- 0.0707} + {0.5502{\mathbb{i}}}} & {0.3560 - {0.0416{\mathbb{i}}}} \\ {0.1649 - {0.4171{\mathbb{i}}}} & {{- 0.1172} - {0.1069{\mathbb{i}}}} & {{- 0.7392} - {0.4413{\mathbb{i}}}} \end{matrix}$ ${V\; 3\left( {\text{:},\text{:},45} \right)} = \begin{matrix} {0.2821 - {0.1279{\mathbb{i}}}} & {{- 0.5649} + {0.3170{\mathbb{i}}}} & {0.3752 + {0.3388{\mathbb{i}}}} \\ {{- 0.0004} - {0.0348{\mathbb{i}}}} & {0.5468 - {0.3850{\mathbb{i}}}} & {0.6392 + {0.3007{\mathbb{i}}}} \\ {{- 0.2923} + {0.0831{\mathbb{i}}}} & {{- 0.2695} + {0.2250{\mathbb{i}}}} & {0.4521 - {0.0693{\mathbb{i}}}} \\ {{- 0.0749} - {0.8971{\mathbb{i}}}} & {{- 0.0004} - {0.0994{\mathbb{i}}}} & {{- 0.1722} - {0.0816{\mathbb{i}}}} \end{matrix}$ ${V\; 3\left( {\text{:},\text{:},46} \right)} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 \\ {{- 0.3536} + {0.3536{\mathbb{i}}}} & {{- 0.3536} - {0.3536{\mathbb{i}}}} & {0.3536 - {0.3536{\mathbb{i}}}} \\ {0 - {0.5000{\mathbb{i}}}} & {0 + {0.5000{\mathbb{i}}}} & {0 - {0.5000{\mathbb{i}}}} \\ {0.3536 + {0.3536{\mathbb{i}}}} & {0.3536 - {0.3536{\mathbb{i}}}} & {{- 0.3536} - {0.3536{\mathbb{i}}}} \end{matrix}$ ${V\; 3\left( {\text{:},\text{:},47} \right)} = \begin{matrix} {{- 0.4748} + {0.1205{\mathbb{i}}}} & {{- 0.1361} + {0.1678{\mathbb{i}}}} & {0.6410 + {0.2967{\mathbb{i}}}} \\ {0.2258 + {0.2637{\mathbb{i}}}} & {{- 0.0187} - {0.5564{\mathbb{i}}}} & {0.4803 + {0.2985{\mathbb{i}}}} \\ {{- 0.265} + {0.3289{\mathbb{i}}}} & {{- 0.3000} + {0.5816{\mathbb{i}}}} & {{- 0.0870} + {0.0961{\mathbb{i}}}} \\ {{- 0.2798} + {0.6188{\mathbb{i}}}} & {{- 0.1970} - {0.4199{\mathbb{i}}}} & {{- 0.3841} - {0.1305{\mathbb{i}}}} \end{matrix}$

${V\; 3\left( {\text{:},\text{:},48} \right)} = \begin{matrix} {{- 0.0180} - {0.6214{\mathbb{i}}}} & {{- 0.4511} + {0.0275{\mathbb{i}}}} & {0.2056 + {0.1427{\mathbb{i}}}} \\ {{- 0.0262} + {0.4269{\mathbb{i}}}} & {0.2038 - {0.1591{\mathbb{i}}}} & {{- 0.3709} - {0.0966{\mathbb{i}}}} \\ {0.1087 + {0.3811{\mathbb{i}}}} & {{- 0.7745} + {0.3487{\mathbb{i}}}} & {{- 0.2829} - {0.1551{\mathbb{i}}}} \\ {{- 0.5069} + {0.1291{\mathbb{i}}}} & {{- 0.0765} - {0.0408{\mathbb{i}}}} & {{- 0.2182} + {0.7993{\mathbb{i}}}} \end{matrix}$ ${V\; 3\left( {\text{:},\text{:},49} \right)} = \begin{matrix} {0.0906 - {0.2217{\mathbb{i}}}} & {0.1089 - {0.3913{\mathbb{i}}}} & {0.6994 + {0.1962{\mathbb{i}}}} \\ {{- 0.3471} + {0.3239{\mathbb{i}}}} & {{- 0.5370} - {0.3087{\mathbb{i}}}} & {{- 0.3246} + {0.1888{\mathbb{i}}}} \\ {{- 0.2568} + {0.4202{\mathbb{i}}}} & {0.3284 + {0.4249{\mathbb{i}}}} & {0.2167 - {0.4149{\mathbb{i}}}} \\ {0.6329 - {0.2724{\mathbb{i}}}} & {{- 0.3826} + {0.1289{\mathbb{i}}}} & {{- 0.0824} - {0.3247{\mathbb{i}}}} \end{matrix}$ ${V\; 3\left( {\text{:},\text{:},50} \right)} = \begin{matrix} {{- 0.2542} - {0.0969{\mathbb{i}}}} & {0.6056 - {0.2040{\mathbb{i}}}} & {{- 0.2829} - {0.4240{\mathbb{i}}}} \\ {0.1965 + {0.0675{\mathbb{i}}}} & {{- 0.4536} - {0.0227{\mathbb{i}}}} & {0.2347 - {0.7163{\mathbb{i}}}} \\ {0.8534 + {0.0091{\mathbb{i}}}} & {0.3944 + {0.0214{\mathbb{i}}}} & {0.0241 + {0.1682{\mathbb{i}}}} \\ {0.3805 + {0.0988{\mathbb{i}}}} & {{- 0.1788} - {0.4443{\mathbb{i}}}} & {{- 0.2841} - {0.2500{\mathbb{i}}}} \end{matrix}$ ${V\; 3\left( {\text{:},\text{:},51} \right)} = \begin{matrix} {{- 0.0310} + {0.3127{\mathbb{i}}}} & {{- 0.4192} + {0.2451{\mathbb{i}}}} & {{- 0.6324} + {0.1246{\mathbb{i}}}} \\ {{- 0.4944} - {0.1906{\mathbb{i}}}} & {0.1428 + {0.6550{\mathbb{i}}}} & {{- 0.0352} + {0.1364{\mathbb{i}}}} \\ {0.0394 + {0.1700{\mathbb{i}}}} & {0.4123 + {0.2162{\mathbb{i}}}} & {{- 0.4882} - {0.5143{\mathbb{i}}}} \\ {{- 0.1337} - {0.7565{\mathbb{i}}}} & {0.2889 - {0.1209{\mathbb{i}}}} & {{- 0.2467} - {0.0313{\mathbb{i}}}} \end{matrix}$

${V\; 3\left( {\text{:},\text{:},52} \right)} = \begin{matrix} {0.0769 + {0.4899{\mathbb{i}}}} & {0.3574 - {0.2210{\mathbb{i}}}} & {{- 0.2987} - {0.2809{\mathbb{i}}}} \\ {0.0623 - {0.0897{\mathbb{i}}}} & {0.7253 - {0.1454{\mathbb{i}}}} & {0.4881 + {0.4323{\mathbb{i}}}} \\ {{- 0.4402} + {0.1739{\mathbb{i}}}} & {{- 0.0113} + {0.3414{\mathbb{i}}}} & {0.2179 - {0.2281{\mathbb{i}}}} \\ {{- 0.7025} - {0.1568{\mathbb{i}}}} & {0.1428 - {0.3730{\mathbb{i}}}} & {{- 0.4909} + {0.2576{\mathbb{i}}}} \end{matrix}$ ${V\; 3\left( {\text{:},\text{:},53} \right)} = \begin{matrix} {{- 0.1844} + {0.1638{\mathbb{i}}}} & {{- 0.6587} + {0.3087{\mathbb{i}}}} & {{- 0.1539} - {0.3692{\mathbb{i}}}} \\ {0.3389 - {0.3843{\mathbb{i}}}} & {{- 0.2949} + {0.3309{\mathbb{i}}}} & {0.4206 - {0.3378{\mathbb{i}}}} \\ {0.4409 - {0.5467{\mathbb{i}}}} & {{- 0.1139} + {0.1683{\mathbb{i}}}} & {{- 0.1714} + {0.4313{\mathbb{i}}}} \\ {{- 0.1168} - {0.4120{\mathbb{i}}}} & {{- 0.2644} - {0.4039{\mathbb{i}}}} & {{- 0.5775} - {0.0090{\mathbb{i}}}} \end{matrix}$ ${V\; 3\left( {\text{:},\text{:},54} \right)} = \begin{matrix} {{- 0.1461} + {0.1056{\mathbb{i}}}} & {0.2135 + {0.2312{\mathbb{i}}}} & {{- 0.6745} + {0.4757{\mathbb{i}}}} \\ {0.1171 - {0.2010{\mathbb{i}}}} & {0.0162 + {0.6417{\mathbb{i}}}} & {{- 0.1534} + {0.0178{\mathbb{i}}}} \\ {{- 0.9157} - {0.2359{\mathbb{i}}}} & {{- 0.0474} - {0.0483{\mathbb{i}}}} & {0.1159 - {0.0681{\mathbb{i}}}} \\ {{- 0.0670} - {0.1213{\mathbb{i}}}} & {{- 0.6245} - {0.3072{\mathbb{i}}}} & {{- 0.2668} + {0.4534{\mathbb{i}}}} \end{matrix}$ ${V\; 3\left( {\text{:},\text{:},55} \right)} = \begin{matrix} {0.5295 + {0.2225{\mathbb{i}}}} & {{- 0.1373} - {0.3127{\mathbb{i}}}} & {{- 0.4137} + {0.3638{\mathbb{i}}}} \\ {0.0632 + {0.0938{\mathbb{i}}}} & {0.1078 + {0.6964{\mathbb{i}}}} & {{- 0.4209} - {0.2518{\mathbb{i}`}}} \\ {{- 0.3442} + {0.5791{\mathbb{i}}}} & {{- 0.4500} - {0.0778{\mathbb{i}}}} & {{- 0.1048} + {0.2769{\mathbb{i}}}} \\ {0.1494 - {0.4256{\mathbb{i}}}} & {{- 0.4187} - {0.0536{\mathbb{i}}}} & {{- 0.5783} - {0.1840{\mathbb{i}}}} \end{matrix}$

${V\; 3\left( {\text{:},\text{:},56} \right)} = \begin{matrix} {{- 0.2534} + {0.0743{\mathbb{i}}}} & {{- 0.3840} + {0.2583{\mathbb{i}}}} & {{- 0.2059} - {0.7267{\mathbb{i}}}} \\ {{- 0.0802} + {0.0767{\mathbb{i}}}} & {0.0963 + {0.7666{\mathbb{i}}}} & {0.2344 + {0.2101{\mathbb{i}}}} \\ {{- 0.3092} + {0.6079{\mathbb{i}}}} & {{- 0.0022} - {0.4149{\mathbb{i}}}} & {{- 0.2681} + {0.1439{\mathbb{i}}}} \\ {{- 0.6565} - {0.1479{\mathbb{i}}}} & {{- 0.1295} + {0.0072{\mathbb{i}}}} & {0.1480 + {0.4646{\mathbb{i}}}} \end{matrix}$ ${V\; 3\left( {\text{:},\text{:},57} \right)} = \begin{matrix} {{- 0.4933} + {0.2493{\mathbb{i}}}} & {{- 0.1093} - {0.6284{\mathbb{i}}}} & {{- 0.1925} - {0.0244{\mathbb{i}}}} \\ {{- 0.0119} - {0.1919{\mathbb{i}}}} & {{- 0.4573} - {0.0414{\mathbb{i}}}} & {0.6333 + {0.3179{\mathbb{i}}}} \\ {0.2792 + {0.2603{\mathbb{i}}}} & {{- 0.2294} - {0.3536{\mathbb{i}}}} & {0.0703 - {0.6494{\mathbb{i}}}} \\ {{- 0.5616} - {0.4432{\mathbb{i}}}} & {{- 0.4520} + {0.0189{\mathbb{i}}}} & {{- 0.1228} - {0.1357{\mathbb{i}}}} \end{matrix}$ ${V\; 3\left( {\text{:},\text{:},58} \right)} = \begin{matrix} {0.5987 + {0.4276{\mathbb{i}}}} & {0.1110 + {0.0636`{\mathbb{i}}}} & {0.3849 + {0.0409{\mathbb{i}}}} \\ {{- 0.2253} + {0.0427{\mathbb{i}}}} & {0.9463 - {0.0868{\mathbb{i}}}} & {{- 0.0474} + {0.0990{\mathbb{i}}}} \\ {{- 0.1122} + {0.5394{\mathbb{i}}}} & {0.0270 + {0.1660{\mathbb{i}}}} & {0.0832 - {0.7139{\mathbb{i}}}} \\ {0.2949 - {0.1252{\mathbb{i}}}} & {0.1828 + {0.1374{\mathbb{i}}}} & {0.5310 + {0.1992{\mathbb{i}}}} \end{matrix}$ ${V\; 3\left( {\text{:},\text{:},59} \right)} = \begin{matrix} {{- 0.1299} - {0.1774{\mathbb{i}}}} & {0.0635 + {0.6372{\mathbb{i}}}} & {{- 0.1454} - {0.5201{\mathbb{i}}}} \\ {{- 0.7371} - {0.3712{\mathbb{i}}}} & {0.1343 - {0.0486{\mathbb{i}}}} & {0.0889 - {0.2013{\mathbb{i}}}} \\ {{- 0.1244} - {0.1207{\mathbb{i}}}} & {{- 0.2344} - {0.6779{\mathbb{i}}}} & {0.0211 - {0.4528{\mathbb{i}}}} \\ {0.4090 - {0.2705{\mathbb{i}}}} & {0.0763 - {0.2220{\mathbb{i}}}} & {{- 0.6105} - {0.2859{\mathbb{i}}}} \end{matrix}$

${V\; 3\left( {\text{:},\text{:},60} \right)} = \begin{matrix} {{- 0.3263} - {0.7161{\mathbb{i}}}} & {{- 0.0074} - {0.1928{\mathbb{i}}}} & {{- 0.3735} + {0.4091{\mathbb{i}}}} \\ {{- 0.0538} + {0.0491{\mathbb{i}}}} & {{- 0.7846} + {0.2480{\mathbb{i}}}} & {{- 0.2825} - {0.2907{\mathbb{i}}}} \\ {0.2749 - {0.2169{\mathbb{i}}}} & {0.0684 + {0.2639{\mathbb{i}}}} & {0.5764 + {0.1916{\mathbb{i}}}} \\ {0.4347 - {0.2526{\mathbb{i}}}} & {{- 0.4391} - {0.1363{\mathbb{i}}}} & {0.2853 + {0.2800{\mathbb{i}}}} \end{matrix}$ ${V\; 3\left( {\text{:},\text{:},61} \right)} = \begin{matrix} {{- 0.3550} + {0.4986{\mathbb{i}}}} & {0.2360 - {0.2295{\mathbb{i}}}} & {0.3696 + {0.3610{\mathbb{i}}}} \\ {{- 0.5979} - {0.4466{\mathbb{i}}}} & {{- 0.1015} - {0.0709{\mathbb{i}}}} & {0.4087 - {0.1032{\mathbb{i}}}} \\ {0.2014 + {0.0225{\mathbb{i}}}} & {{- 0.1794} + {0.3659{\mathbb{i}}}} & {0.6567 + {0.3341{\mathbb{i}}}} \\ {{- 0.1636} - {0.0232{\mathbb{i}}}} & {{- 0.8345} + {0.1176{\mathbb{i}}}} & {{- 0.0448} - {0.1024{\mathbb{i}}}} \end{matrix}$ ${V\; 3\left( {\text{:},\text{:},62} \right)} = \begin{matrix} {0.4315 + {0.2977{\mathbb{i}}}} & {0.5279 - {0.2633{\mathbb{i}}}} & {{- 0.0096} - {0.1863{\mathbb{i}}}} \\ {{- 0.3022} + {0.611{\mathbb{i}}}} & {0.0004 + {0.3129{\mathbb{i}}}} & {0.4461 + {0.2312{\mathbb{i}}}} \\ {{- 0.4238} - {0.3362{\mathbb{i}}}} & {0.4544 + {0.2155{\mathbb{i}}}} & {{- 0.6006} - {0.2328{\mathbb{i}}}} \\ {0.3188 - {0.4857{\mathbb{i}}}} & {0.5285 + {0.1480{\mathbb{i}}}} & {0.2559 + {0.4820{\mathbb{i}}}} \end{matrix}$ ${V\; 3\left( {\text{:},\text{:},63} \right)} = \begin{matrix} {0.6135 - {0.5466{\mathbb{i}}}} & {0.1363 + {0.0107{\mathbb{i}}}} & {{- 0.0265} - {0.2357{\mathbb{i}}}} \\ {{- 0.3336} + {0.1681{\mathbb{i}}}} & {0.3619 + {0.5204{\mathbb{i}}}} & {0.4467 - {0.0956{\mathbb{i}}}} \\ {0.0923 + {0.1646{\mathbb{i}}}} & {{- 0.3085} - {0.5415{\mathbb{i}}}} & {0.4468 + {0.3555{\mathbb{i}}}} \\ {0.2389 + {0.3044{\mathbb{i}}}} & {{- 0.1937} + {0.3920{\mathbb{i}}}} & {{- 0.5692} + {0.2917{\mathbb{i}}}} \end{matrix}$

${V\; 3\left( {\text{:},\text{:},64} \right)} = \begin{matrix} {{- 0.0643} + {0.2882{\mathbb{i}}}} & {0.7408 - {0.0532{\mathbb{i}}}} & {{- 0.0130} - {0.1806{\mathbb{i}}}} \\ {0.1360 - {0.4277{\mathbb{i}}}} & {0.4029 + {0.4160{\mathbb{i}}}} & {{- 0.4410} - {0.0947{\mathbb{i}}}} \\ {{- 0.4682} + {0.3860{\mathbb{i}}}} & {0.0628 + {0.1030{\mathbb{i}}}} & {0.4010 - {0.2857{\mathbb{i}}}} \\ {{- 0.2986} - {0.5040{\mathbb{i}}}} & {{- 0.0741} - {0.3049{\mathbb{i}}}} & {{- 0.0166} - {0.7218{\mathbb{i}}}} \end{matrix}$

-   -   Final Rank 4 Codebook:

${V\; 4\left( {\text{:},\text{:},1} \right)} = \begin{matrix} 0.5000 & 0.5000 & {- 0.5000} & {- 0.5000} \\ 0.5000 & {- 0.5000} & {- 0.5000} & 0.5000 \\ 0.5000 & 0.5000 & 0.5000 & 0.5000 \\ 0.5000 & {- 0.5000} & 0.5000 & {- 0.5000} \end{matrix}$ ${V\; 4\left( {\text{:},\text{:},2} \right)} = \begin{matrix} 0.5000 & 0.5000 & {- 0.5000} & {- 0.5000} \\ {0 + {0.5000{\mathbb{i}}}} & {0 - {0.5000{\mathbb{i}}}} & {0 - {0.5000{\mathbb{i}}}} & {0 + {0.5000{\mathbb{i}}}} \\ 0.5000 & 0.5000 & 0.5000 & 0.5000 \\ {0 + {0.5000{\mathbb{i}}}} & {0 - {0.5000{\mathbb{i}}}} & {0 + {0.5000{\mathbb{i}}}} & {0 - {0.5000{\mathbb{i}}}} \end{matrix}$ ${V\; 4\left( {\text{:},\text{:},3} \right)} = \begin{matrix} 0.5000 & 0.5000 & {- 0.5000} & {- 0.5000} \\ 0.5000 & {- 0.5000} & {0 - {0.5000{\mathbb{i}}}} & {0 + {0.5000{\mathbb{i}}}} \\ 0.5000 & 0.5000 & 0.5000 & 0.5000 \\ 0.5000 & {- 0.5000} & {0 + {0.5000{\mathbb{i}}}} & {0 - {0.5000{\mathbb{i}}}} \end{matrix}$ ${V\; 4\left( {\text{:},\text{:},4} \right)} = \begin{matrix} 0.5000 & 0.5000 & {- 0.5000} & {- 0.5000} \\ {0 + {0.5000{\mathbb{i}}}} & {0 - {0.5000{\mathbb{i}}}} & {- 0.5000} & 0.5000 \\ 0.5000 & 0.5000 & 0.5000 & 0.5000 \\ {0 + {0.5000{\mathbb{i}}}} & {0 - {0.5000{\mathbb{i}}}} & 0.5000 & {- 0.5000} \end{matrix}$

$\mspace{79mu}{{V\; 4\left( {\text{:},\text{:},5} \right)} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 & 0.5000 \\ 0.5000 & {0 + {0.5000{\mathbb{i}}}} & {- 0.5000} & {0 - {0.5000{\mathbb{i}}}} \\ 0.5000 & {- 0.5000} & 0.5000 & {- 0.5000} \\ {- 0.5000} & {0 + {0.5000{\mathbb{i}}}} & 0.5000 & {0 - {0.5000{\mathbb{i}}}} \end{matrix}}$ ${V\; 4\left( {\text{:},\text{:},6} \right)} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 & 0.5000 \\ {0.3536 + {0.3536{\mathbb{i}}}} & {{- 0.3536} + {0.3536{\mathbb{i}}}} & {{- 0.3536} - {0.3536{\mathbb{i}}}} & {0.3536 - {0.3536{\mathbb{i}}}} \\ {0 + {0.5000{\mathbb{i}}}} & {0 - {0.5000{\mathbb{i}}}} & {0 + {0.5000{\mathbb{i}}}} & {0 - {0.5000{\mathbb{i}}}} \\ {{- 0.3536} + {0.3536{\mathbb{i}}}} & {0.3536 + {0.3536{\mathbb{i}}}} & {0.3536 - {0.3536{\mathbb{i}}}} & {{- 0.3536} - {0.3536{\mathbb{i}}}} \end{matrix}$ ${V\; 4\left( {\text{:},\text{:},7} \right)} = \begin{matrix} {{- 0.2573} + {0.5267{\mathbb{i}}}} & {{- 0.4516} - {0.0164{\mathbb{i}}}} & {{- 0.1786} + {0.5269{\mathbb{i}}}} & {{- 0.2985} - {0.2313{\mathbb{i}}}} \\ {0.4306 + {0.2510{\mathbb{i}}}} & {{- 0.3631} - {0.4030{\mathbb{i}}}} & {{- 0.3685} - {0.1544{\mathbb{i}}}} & {0.1966 + {0.5090{\mathbb{i}}}} \\ {0.4306 + {0.2510{\mathbb{i}}}} & {0.4237 + {0.2733{\mathbb{i}}}} & {{- 0.5798} - {0.0933{\mathbb{i}}}} & {{- 0.1123} - {0.3741{\mathbb{i}}}} \\ {{- 0.3995} - {0.0009{\mathbb{i}}}} & {0.2522 + {0.4286{\mathbb{i}}}} & {{- 0.3685} + {0.2240{\mathbb{i}}}} & {0.0615 + {0.6351{\mathbb{i}}}} \end{matrix}$ ${V\; 4\left( {\text{:},\text{:},8} \right)} = \begin{matrix} {0.4182 - {0.2060{\mathbb{i}}}} & {0.7071 - {0.3537{\mathbb{i}}}} & {{- 0.0971} + {0.0208{\mathbb{i}}}} & {{- 0.2518} - {0.2904{\mathbb{i}}}} \\ {0.2693 + {0.5849{\mathbb{i}}}} & {{- 0.1549} - {0.4387{\mathbb{i}}}} & {{- 0.4646} - {0.3323{\mathbb{i}}}} & {0.1746 + {0.1101{\mathbb{i}}}} \\ {{- 0.3088} + {0.1985{\mathbb{i}}}} & {0.3107 - {0.1994{\mathbb{i}}}} & {{- 0.1616} + {0.\; 7129{\mathbb{i}}}} & {0.4094 + {0.1641{\mathbb{i}}}} \\ {{- 0.1743} + {0.4504{\mathbb{i}}}} & {{- 0.1065} + {0.1039{\mathbb{i}}}} & {{- 0.1624} + {0.3212{\mathbb{i}}}} & {{- 0.7503} - {0.2284{\mathbb{i}}}} \end{matrix}$

${V\; 4\left( {:{,{:{,9}}}} \right)} = \begin{matrix} {{- 0.1228} + {0.0831{\mathbb{i}}}} & {0.1911 + {0.1844{\mathbb{i}}}} & {{- 0.2742} - {0.7346{\mathbb{i}}}} & {0.4333 - {0.3239{\mathbb{i}}}} \\ {0.4892 - {0.2949{\mathbb{i}}}} & {{- 0.6076} - {0.2220{\mathbb{i}}}} & {0.1147 - {0.2232{\mathbb{i}}}} & {{- 0.0716} - {0.4327{\mathbb{i}}}} \\ {0.4754 + {0.1031{\mathbb{i}}}} & {0.0935 - {0.1294{\mathbb{i}}}} & {{- 0.2568} - {0.4393{\mathbb{i}}}} & {{- 0.3198} + {0.6137{\mathbb{i}}}} \\ {{- 0.3409} - {0.5467{\mathbb{i}}}} & {0.3520 - {0.6014{\mathbb{i}}}} & {0.1308 - {0.2148{\mathbb{i}}}} & {{- 0.1887} - {0.0218{\mathbb{i}}}} \end{matrix}$ ${V\; 4\left( {:{,{:{,10}}}} \right)} = \begin{matrix} {0.1663 + {0.6240{\mathbb{i}}}} & {{- 0.4390} - {0.0177{\mathbb{i}}}} & {0.1890 + {0.1915{\mathbb{i}}}} & {0.5399 - {0.1614{\mathbb{i}}}} \\ {0.0064 + {0.1030{\mathbb{i}}}} & {{- 0.2573} - {0.1623{\mathbb{i}}}} & {0.5097 - {0.7079{\mathbb{i}}}} & {{- 0.1643} + {0.3299{\mathbb{i}}}} \\ {{- 0.3928} + {0.4752{\mathbb{i}}}} & {0.4550 + {0.6115{\mathbb{i}}}} & {0.1352 - {0.1381{\mathbb{i}}}} & {{- 0.0396} - {0.0100{\mathbb{i}}}} \\ {{- 0.4732} - {0.0316{\mathbb{i}}}} & {0.0477 - {0.3623{\mathbb{i}}}} & {{- 0.0601} - {03545{\mathbb{i}}}} & {0.1275 - {0.7271{\mathbb{i}}}} \end{matrix}$ ${V\; 4\left( {:{,{:{,11}}}} \right)} = \begin{matrix} {0.1228 - {0.0831{\mathbb{i}}}} & {{{- 0.0832}{\mathbb{i}}} + {0.1719{\mathbb{i}}}} & {0.1018 + {0.2933{\mathbb{i}}}} & {0.3119 - {0.2093{\mathbb{i}}}} \\ {{- 0.4754} - {0.1031{\mathbb{i}}}} & {0.0004 - {0.0643{\mathbb{i}}}} & {0.6544 - {0.1493{\mathbb{i}}}} & {0.3891 + {0.3966{\mathbb{i}}}} \\ {{- 0.4892} + {0.2949{\mathbb{i}}}} & {0.0542 + {0.3715{\mathbb{i}}}} & {{- 0.5068} + {0.3662{\mathbb{i}}}} & {0.2327 + {0.2963{\mathbb{i}}}} \\ {0.3409 + {0.5467{\mathbb{i}}}} & {{- 0.2612} + {0.2147{\mathbb{i}}}} & {0.1621 - {0.1894{\mathbb{i}}}} & {{- 0.3150} + {0.5560{\mathbb{i}}}} \end{matrix}$ ${V\; 4\left( {:{,{:12}}} \right)} = \begin{matrix} {0.4119 - {0.2376{\mathbb{i}}}} & {0.2307 - {0.3523{\mathbb{i}}}} & {{- 0.3946} + {0.2222{\mathbb{i}}}} & {0.3873 + {0.4914{\mathbb{i}}}} \\ {0.0073 - {0.7328{\mathbb{i}}}} & {{- 0.1995} - {0.0992{\mathbb{i}}}} & {0.1664 - {0.1501{\mathbb{i}}}} & {{- 0.5320} + {0.2829{\mathbb{i}}}} \\ {0.3791 + {0.1546{\mathbb{i}}}} & {{- 0.6261} + {0.4891{\mathbb{i}}}} & {0.1146 + {0.2907{\mathbb{i}}}} & {0.0580 + {0.3166{\mathbb{i}}}} \\ {0.2445 - {0.0972{\mathbb{i}}}} & {{- 0.3749} - {0.0349{\mathbb{i}}}} & {{- 0.2411} - {0.7674{\mathbb{i}}}} & {0.3196 - {0.1994{\mathbb{i}}}} \end{matrix}$

${V\; 4\left( {\text{:},\text{:},13} \right)} = \begin{matrix} {{- 0.7073} - {0.3349{\mathbb{i}}}} & {{- 0.3380} - {0.1860{\mathbb{i}}}} & {0.2936 - {0.0670{\mathbb{i}}}} & {0.1233 + {0.3645{\mathbb{i}}}} \\ {{- 0.2125} + {0.3788{\mathbb{i}}}} & {{- 0.2536} - {0.2825{\mathbb{i}}}} & {{- 0.4027} + {0.1913{\mathbb{i}}}} & {{- 0.6695} + {0.1424{\mathbb{i}}}} \\ {{- 0.2125} + {0.3788{\mathbb{i}}}} & {0.6905 - {0.2237{\mathbb{i}}}} & {0.4699 + {0.1884{\mathbb{i}}}} & {{- 0.0830} + {0.1462{\mathbb{i}}}} \\ {0.0779 + {0.0648{\mathbb{i}}}} & {{- 0.3771} + {0.1950{\mathbb{i}}}} & {0.6721 + {0.0505{\mathbb{i}}}} & {{- 0.4095} - {0.4330{\mathbb{i}}}} \end{matrix}$ ${V\; 4\left( {\text{:},\text{:},14} \right)} = \begin{matrix} {{- 0.0318} - {0.3722{\mathbb{i}}}} & {0.5870 + {0.5837{\mathbb{i}}}} & {0.1694 - {03686{\mathbb{i}}}} & {{- 0.1020} - {0.0171{\mathbb{i}}}} \\ {{- 0.0512} - {0.1869{\mathbb{i}}}} & {0.1353 - {0.0302{\mathbb{i}}}} & {0.2036 + {0.5782{\mathbb{i}}}} & {{- 0.5358} + {0.5295{\mathbb{i}}}} \\ {0.5269 + {0.1995{\mathbb{i}}}} & {0.1672 - {0.4033{\mathbb{i}}}} & {0.4862 - {0.2843{\mathbb{i}}}} & {{- 0.3698} - {0.1949{\mathbb{i}}}} \\ {0.3031 - {0.6431{\mathbb{i}}}} & {0.1619 - {0.2805{\mathbb{i}}}} & {0.0806 + {0.3688{\mathbb{i}}}} & {0.4237 - {0.2601{\mathbb{i}}}} \end{matrix}$ ${V\; 4\left( {\text{:},\text{:},15} \right)} = \begin{matrix} {0.0318 + {0.3722{\mathbb{i}}}} & {0.2910 + {0.0846{\mathbb{i}}}} & {0.0464 + {0.0232{\mathbb{i}}}} & {{- 0.8617} + {0.1530{\mathbb{i}}}} \\ {0.1869 - {0.0512{\mathbb{i}}}} & {{- 0.5838} - {0.6826{\mathbb{i}}}} & {0.1164 + {0.0920{\mathbb{i}}}} & {{- 0.3062} - {0.1997{\mathbb{i}}}} \\ {0.5269 + {0.1995{\mathbb{i}}}} & {{- 0.1084} + {0.1034{\mathbb{i}}}} & {{- 0.6815} - {0.4049{\mathbb{i}}}} & {{- 0.0000} - {0.1781{\mathbb{i}}}} \\ {{- 0.6431} - {0.3031{\mathbb{i}}}} & {0.0232 - {0.2799{\mathbb{i}}}} & {{- 0.4827} - {0.3376{\mathbb{i}}}} & {{- 0.1746} + {0.1956{\mathbb{i}}}} \end{matrix}$ ${V\; 4\left( {\text{:},\text{:},16} \right)} = \begin{matrix} {{- 0.1228} + {0.0831{\mathbb{i}}}} & {0.4864 + {0.1062{\mathbb{i}}}} & {{- 0.5138} + {0.1606{\mathbb{i}}}} & {0.6091 + {0.2633{\mathbb{i}}}} \\ {{- 0.1031} + {0.4754{\mathbb{i}}}} & {{- 0.3579} + {0.0709{\mathbb{i}}}} & {0.2824 - {0.4495{\mathbb{i}}}} & {0.5834 - {0.0898{\mathbb{i}}}} \\ {{- 0.4892} + {0.2949{\mathbb{i}}}} & {0.3743 - {0.4217{\mathbb{i}}}} & {{- 0.1744} - {0.4396{\mathbb{i}}}} & {{- 0.3251} - {0.1626{\mathbb{i}}}} \\ {{- 0.5467} + {0.3409{\mathbb{i}}}} & {{- 0.2104} + {0.5067{\mathbb{i}}}} & {{- 0.1120} + {0.4384{\mathbb{i}}}} & {{- 0.2030} - {0.1946{\mathbb{i}}}} \end{matrix}$

${V\; 4\left( {\text{:},\text{:},17} \right)} = \begin{matrix} {{- 0.4119} + {0.2376{\mathbb{i}}}} & {{- 0.3475} + {0.0349{\mathbb{i}}}} & {{- 0.5492} + {0.0805{\mathbb{i}}}} & {0.4831 - {0.3324{\mathbb{i}}}} \\ {0.7328 + {0.0073{\mathbb{i}}}} & {{- 0.3183} + {0.2743{\mathbb{i}}}} & {0.0020 - {0.0107{\mathbb{i}}}} & {0.5026 + {0.1833{\mathbb{i}}}} \\ {0.3791 + {0.1546{\mathbb{i}}}} & {0.3910 - {0.0682{\mathbb{i}}}} & {{- 0.6621} - {0.4101{\mathbb{i}}}} & {{- 0.2328} - {0.1188{\mathbb{i}}}} \\ {{- 0.0972} - {0.2445{\mathbb{i}}}} & {{- 0.5729} + {0.4644{\mathbb{i}}}} & {{- 0.2236} - {0.1878{\mathbb{i}}}} & {{- 0.5285} + {0.1492{\mathbb{i}}}} \end{matrix}$ ${V\; 4\left( {\text{:},\text{:},18} \right)} = \begin{matrix} {0.7073 + {0.3349{\mathbb{i}}}} & {0.3580 + {0.4230{\mathbb{i}}}} & {0.2367 + {0.0128{\mathbb{i}}}} & {0.0222 - {0.1544{\mathbb{i}}}} \\ {0.3788 + {0.2125{\mathbb{i}}}} & {{- 0.5016} - {0.3133{\mathbb{i}}}} & {0.2038 - {0.2663{\mathbb{i}}}} & {{- 0.4319} + {0.4031{\mathbb{i}}}} \\ {{- 0.2125} + {0.3788{\mathbb{i}}}} & {0.4516 - {0.0120{\mathbb{i}}}} & {{- 0.3421} - {0.6404{\mathbb{i}}}} & {{- 0.0186} + {0.2825{\mathbb{i}}}} \\ {{- 0.0648} + {0.0779{\mathbb{i}}}} & {0.2277 - {0.2953{\mathbb{i}}}} & {0.5298 + {0.1531{\mathbb{i}}}} & {0.5491 + {0.4950{\mathbb{i}}}} \end{matrix}$ ${V\; 4\left( {\text{:},\text{:},19} \right)} = \begin{matrix} {{- 0.1663} - {0.6240{\mathbb{i}}}} & {{- 0.0285} - {0.3000{\mathbb{i}}}} & {0.2482 - {0.1368{\mathbb{i}}}} & {{- 0.3683} + {0.5255{\mathbb{i}}}} \\ {{- 0.1030} + {0.0064{\mathbb{i}}}} & {{- 0.0146} + {0.4747{\mathbb{i}}}} & {0.8444 + {0.1708{\mathbb{i}}}} & {0.1466 - {0.0063{\mathbb{i}}}} \\ {{- 0.3928} + {0.4752{\mathbb{i}}}} & {0.1862 + {0.2754{\mathbb{i}}}} & {{- 0.1088} + {0.1084{\mathbb{i}}}} & {{- 0.6599} + {0.2244{\mathbb{i}}}} \\ {{- 0.0316} + {0.4372{\mathbb{i}}}} & {{- 0.4436} - {0.6134{\mathbb{i}}}} & {0.3812 - {0.0925{\mathbb{i}}}} & {{- 0.1957} - {0.2063{\mathbb{i}}}} \end{matrix}$ ${V\; 4\left( {\text{:},\text{:},20} \right)} = \begin{matrix} {0.2573 - {0.5267{\mathbb{i}}}} & {0.4288 + {0.0624{\mathbb{i}}}} & {{- 0.4668} + {0.1127{\mathbb{i}}}} & {{- 0.4182} + {0.2512{\mathbb{i}}}} \\ {0.2510 - {0.4306{\mathbb{i}}}} & {{- 0.2155} - {0.5900{\mathbb{i}}}} & {0.4068 - {0.1439{\mathbb{i}}}} & {{- 0.2850} - {0.29955{\mathbb{i}}}} \\ {0.4306 + {0.2510{\mathbb{i}}}} & {0.0071 - {0.1770{\mathbb{i}}}} & {0.2316 - {0.2993{\mathbb{i}}}} & {0.0461 + {0.7582{\mathbb{i}}}} \\ {0.0009 - {0.3995{\mathbb{i}}}} & {0.1913 + {0.5913{\mathbb{i}}}} & {0.6629 + {0.0244{\mathbb{i}}}} & {0.0964 + {0.0693{\mathbb{i}}}} \end{matrix}$

${V\; 4\left( {:{,{:21}}} \right)} = \begin{matrix} {{- 0.4182} + {0.2060{\mathbb{i}}}} & {{- 0.0368} - {0.0444{\mathbb{i}}}} & {{- 0.1903} + {0.2671{\mathbb{i}}}} & {0.4025 - {0.7140{\mathbb{i}}}} \\ {{- 0.5849} + {0.2693{\mathbb{i}}}} & {0.0931 + {0.1023{\mathbb{i}}}} & {0.3098 - {0.6795{\mathbb{i}}}} & {{- 0.0892} + {0.0221{\mathbb{i}}}} \\ {{- 0.3088} + {0.1985{\mathbb{i}}}} & {0.6755 + {0.2582{\mathbb{i}}}} & {{- 0.0591} + {04709{\mathbb{i}}}} & {{- 0.0849} + {0.3313{\mathbb{i}}}} \\ {0.4504 + {0.1743{\mathbb{i}}}} & {01401 + {0.6595{\mathbb{i}}}} & {03082 - {0.1202{\mathbb{i}}}} & {0.4248 - {0.1494{\mathbb{i}}}} \end{matrix}$ ${V\; 4\left( {:{,{:{,22}}}} \right)} = \begin{matrix} {0.1228 - {0.0831{\mathbb{i}}}} & {0.5267 + {0.1302{\mathbb{i}}}} & {{- 0.3106} - {0.1437{\mathbb{i}}}} & {0.7407 - {0.1338{\mathbb{i}}}} \\ {{- 0.2949} - {0.4892{\mathbb{i}}}} & {{- 0.3082} + {0.1349{\mathbb{i}}}} & {{- 0.4197} - {0.5805{\mathbb{i}}}} & {{- 0.1306} - {0.1743{\mathbb{i}}}} \\ {0.4754 + {0.1031{\mathbb{i}}}} & {{- 0.5798} - {0.2378{\mathbb{i}}}} & {{- 0.5074} + {0.1972{\mathbb{i}}}} & {0.2367 + {0.1351{\mathbb{i}}}} \\ {0.5467 - {0.3409{\mathbb{i}}}} & {0.4428 + {0.0{.607}{\mathbb{i}}}} & {{- 0.2678} + {0.0417{\mathbb{i}}}} & {{- 0.5235} + {0.1939{\mathbb{i}}}} \end{matrix}$ ${V\; 4\left( {:{,{:{,23}}}} \right)} = \begin{matrix} {{- 0.6051} + {0.1790{\mathbb{i}}}} & {{- 0.2867} + {0.4568{\mathbb{i}}}} & {{- 0.0929} + {0.0656{\mathbb{i}}}} & {{- 0.1384} - {0.5281{\mathbb{i}}}} \\ {0.3147 + {0.1351{\mathbb{i}}}} & {{- 0.3377} + {0.4028{\mathbb{i}}}} & {0.6771 - {0.3179{\mathbb{i}}}} & {0.2166 + {0.0016{\mathbb{i}}}} \\ {0.3147 + {0.1351{\mathbb{i}}}} & {{- 0.4367} - {0.4408{\mathbb{i}}}} & {{- 0.1123} - {0.3181{\mathbb{i}}}} & {{- 0.5156} - {0.3436{\mathbb{i}}}} \\ {{- 0.1801} - {0.5787{\mathbb{i}}}} & {0.2118 - {0.0548{\mathbb{i}}}} & {0.5502 + {0.1048{\mathbb{i}}}} & {{- 0.5105} - {0.1021{\mathbb{i}}}} \end{matrix}$ ${V\; 4\left( {:{,{:{,24}}}} \right)} = \begin{matrix} {0.0704 + {0.1417{\mathbb{i}}}} & {0.4489 - {0.4294{\mathbb{i}}}} & {0.4657 + {0.4116{\mathbb{i}}}} & {0.2255 + {0.3897{\mathbb{i}}}} \\ {0.1534 + {0.7008{\mathbb{i}}}} & {0.0676 - {0.2514{\mathbb{i}}}} & {{- 0.5317} - {0.2639{\mathbb{i}}}} & {0.1962 + {0.1631{\mathbb{i}}}} \\ {{- 0.4248} + {0.3144{\mathbb{i}}}} & {{- 0.0562} - {03389{\mathbb{i}}}} & {{- 0.0047} + {0.3844{\mathbb{i}}}} & {{- 0.1563} - {0.6561{\mathbb{i}}}} \\ {{- 0.4053} + {0.1292{\mathbb{i}}}} & {{- 0.6352} + {0.1576{\mathbb{i}}}} & {{- 0.0225} + {0.3362{\mathbb{i}}}} & {0.3199 + {0.4182{\mathbb{i}}}} \end{matrix}$

${V\; 4\left( {:{,{:{,25}}}} \right)} = \begin{matrix} {0.5141 + {0.2763{\mathbb{i}}}} & {0.0224 - {0.0857{\mathbb{i}}}} & {{- 0.3110} + {0.4440{\mathbb{i}}}} & {0.3131 - {0.5095{\mathbb{i}}}} \\ {0.2189 + {0.1095{\mathbb{i}}}} & {0.4370 + {0.0749{\mathbb{i}}}} & {0.5479 + {0.3445{\mathbb{i}}}} & {0.3368 + {0.4595{\mathbb{i}}}} \\ {0.2769 - {0.3593{\mathbb{i}}}} & {0.0927 - {0.6134{\mathbb{i}}}} & {{- 0.4211} + {0.0796{\mathbb{i}}}} & {{- 0.0820} + {0.4679{\mathbb{i}}}} \\ {{- 0.6111} - {0.1423{\mathbb{i}}}} & {0.6031 - {0.2167{\mathbb{i}}}} & {{- 0.1108} + {0.3020{\mathbb{i}}}} & {{- 0.0056} - {0.3033{\mathbb{i}}}} \end{matrix}$ ${V\; 4\left( {:{,{:{,26}}}} \right)} = \begin{matrix} {{- 0.6051} + {0.1790{\mathbb{i}}}} & {{- 0.2021} + {0.2515{\mathbb{i}}}} & {{- 0.1048} + {0.3295{\mathbb{i}}}} & {0.4277 + {0.4419{\mathbb{i}}}} \\ {{- 0.1351} + {0.3147{\mathbb{i}}}} & {{- 0.2703} - {0.8413{\mathbb{i}}}} & {0.2200 + {0.2221{\mathbb{i}}}} & {{- 0.0633} - {0.0093{\mathbb{i}}}} \\ {{- 0.3147} - {0.1351{\mathbb{i}}}} & {{- 0.3054} - {0.0715{\mathbb{i}}}} & {{- 0.4718} - {0.1404{\mathbb{i}}}} & {0.2258 - {0.7007{\mathbb{i}}}} \\ {{- 0.5787} + {0.1801{\mathbb{i}}}} & {0.1019 + {0.0787{\mathbb{i}}}} & {0.3731 - {0.6334{\mathbb{i}}}} & {{- 0.2723} - {0.0393{\mathbb{i}}}} \end{matrix}$ ${V\; 4\left( {:{,{:{,27}}}} \right)} = \begin{matrix} {0.2250 + {0.4308{\mathbb{i}}}} & {{- 0.0056} + {0.0131{\mathbb{i}}}} & {0.0379 - {0.7861{\mathbb{i}}}} & {0.2371 - {0.2967{\mathbb{i}}}} \\ {0.3732 - {0.4108{\mathbb{i}}}} & {0.0211 + {0.5772{\mathbb{i}}}} & {0.4481 - {0.2070{\mathbb{i}}}} & {0.0255 - {0.3378{\mathbb{i}}}} \\ {0.5913 + {0.2190{\mathbb{i}}}} & {0.6155 - {0.2919{\mathbb{i}}}} & {0.1617 + {0.2064{\mathbb{i}}}} & {{- 0.2637} - {0.0051{\mathbb{i}}}} \\ {{- 0.2387} - {0.0328{\mathbb{i}}}} & {0.3534 - {0.2780{\mathbb{i}}}} & {0.2588 + {0.0366{\mathbb{i}}}} & {0.7632 - {0.2983{\mathbb{i}}}} \end{matrix}$ ${V\; 4\left( {:{,{:{,28}}}} \right)} = \begin{matrix} {0.5141 + {0.2763{\mathbb{i}}}} & {0.0913 - {0.3481{\mathbb{i}}}} & {0.5632 + {0.2802{\mathbb{i}}}} & {{- 0.1951} - {0.3100{\mathbb{i}}}} \\ {{- 0.1095} + {0.2189{\mathbb{i}}}} & {{- 0.5047} + {0.4712{\mathbb{i}}}} & {{- 0.0240} - {0.0899{\mathbb{i}}}} & {{- 0.2776} - {0.6145{\mathbb{i}}}} \\ {{- 0.2769} + {0.3593{\mathbb{i}}}} & {0.0947 - {0.5855{\mathbb{i}}}} & {{- 0.3176} - {0.3672{\mathbb{i}}}} & {{- 0.4474} - {0.0811{\mathbb{i}}}} \\ {{- 0.1423} + {0.6111{\mathbb{i}}}} & {{- 0.0727} + {0.1915{\mathbb{i}}}} & {0.4510 - {0.3955{\mathbb{i}}}} & {0.2488 + {0.3775{\mathbb{i}}}} \end{matrix}$

${V\; 4\left( {:{,{:{,29}}}} \right)} = \begin{matrix} {0.0704 + {0.1417{\mathbb{i}}}} & {0.2602 - {0.3335{\mathbb{i}}}} & {0.1575 - {0.6399{\mathbb{i}}}} & {0.5258 + {0.2920{\mathbb{i}}}} \\ {0.7008 - {0.1534{\mathbb{i}}}} & {0.3267 - {0.3041{\mathbb{i}}}} & {{- 0.2930} + {0.2159{\mathbb{i}}}} & {{- 0.2319} + {0.3158{\mathbb{i}}}} \\ {0.4248 - {0.3144{\mathbb{i}}}} & {0.1781 + {0.6675{\mathbb{i}}}} & {0.1266 - {0.0025{\mathbb{i}}}} & {0.4330 - {0.1995{\mathbb{i}}}} \\ {{- 0.1292} - {0.4053{\mathbb{i}}}} & {{- 0.1799} + {0.3348{\mathbb{i}}}} & {{- 0.3623} - {0.5347{\mathbb{i}}}} & {{- 0.3289} + {0.3863{\mathbb{i}}}} \end{matrix}$ ${V\; 4\left( {:{,{:{,30}}}} \right)} = \begin{matrix} {0.2250 + {0.4308{\mathbb{i}}}} & {{- 0.1449} - {0.3470{\mathbb{i}}}} & {{- 0.3069} - {0.2742{\mathbb{i}}}} & {{- 0.4090} + {0.5345{\mathbb{i}}}} \\ {0.4108 + {0.3732{\mathbb{i}}}} & {{- 0.0692} + {0.7493{\mathbb{i}}}} & {0.0914 + {0.0688{\mathbb{i}}}} & {{- 0.3010} - {0.1486{\mathbb{i}}}} \\ {{- 0.5913} - {0.2190{\mathbb{i}}}} & {{- 0.1504} + {0.4516{\mathbb{i}}}} & {0.1108 - {0.0595{\mathbb{i}}}} & {{- 0.1821} + {0.5717{\mathbb{i}}}} \\ {{- 0.0328} + {0.2387{\mathbb{i}}}} & {0.1073 - {0.2330{\mathbb{i}}}} & {0.4107 + {0.7956{\mathbb{i}}}} & {{- 0.1571} + {0.2230{\mathbb{i}}}} \end{matrix}$ ${V\; 4\left( {:{,{:{,31}}}} \right)} = \begin{matrix} {{- 0.0376} + {0.4566{\mathbb{i}}}} & {{- 0.6780} - {0.3635{\mathbb{i}}}} & {0.0238 - {0.3004{\mathbb{i}}}} & {{- 0.2666} - {0.1906{\mathbb{i}}}} \\ {0.2688 + {0.1481{\mathbb{i}}}} & {0.1160 + {0.0644{\mathbb{i}}}} & {0.7450 + {0.3155{\mathbb{i}}}} & {{- 0.4820} + {0.0363{\mathbb{i}}}} \\ {0.1588 - {0.0744{\mathbb{i}}}} & {{- 0.3301} + {0.2541{\mathbb{i}}}} & {0.4517 - {0.0242{\mathbb{i}}}} & {0.6746 - {0.3688{\mathbb{i}}}} \\ {{- 0.5917} - {0.5613{\mathbb{i}}}} & {{- 0.4384} + {0.1574{\mathbb{i}}}} & {0.1584 + {0.1580{\mathbb{i}}}} & {{- 0.2294} + {0.1232{\mathbb{i}}}} \end{matrix}$ ${V\; 4\left( {:{,{:{,32}}}} \right)} = \begin{matrix} {{- 0.4472} - {0.1208{\mathbb{i}}}} & {0.2383 + {0.0963{\mathbb{i}}}} & {{- 0.0567} - {0.0816{\mathbb{i}}}} & {0.8144 + {0.2149{\mathbb{i}}}} \\ {{- 0.0781} + {0.4936{\mathbb{i}}}} & {0.2810 + {0.0700{\mathbb{i}}}} & {{- 0.5121} + {0.5153{\mathbb{i}}}} & {{- 0.1356} + {0.3467{\mathbb{i}}}} \\ {{- 0.6133} - {0.0191{\mathbb{i}}}} & {0.2857 - {0.4528{\mathbb{i}}}} & {0.4309 + {0.0872{\mathbb{i}}}} & {{- 0.3612} + {0.1144{\mathbb{i}}}} \\ {{- 0.3591} + {0.1738{\mathbb{i}}}} & {{- 0.4779} + {0.5788{\mathbb{i}}}} & {0.3586 + {0.3748{\mathbb{i}}}} & {{- 0.0135} - {0.0906{\mathbb{i}}}} \end{matrix}$

${V\; 4\left( {:{,{:{,33}}}} \right)} = \begin{matrix} {0.4535 + {0.1524{\mathbb{i}}}} & {0.6361 - {0.0379{\mathbb{i}}}} & {0.0303 + {0.1370{\mathbb{i}}}} & {{- 0.3832} + {0.4456{\mathbb{i}}}} \\ {0.1943 - {0.1220{\mathbb{i}}}} & {{- 0.3164} + {0.0577{\mathbb{i}}}} & {0.8288 + {0.2462{\mathbb{i}}}} & {{- 0.2979} - {0.0876{\mathbb{i}}}} \\ {0.7044 - {0.2811{\mathbb{i}}}} & {0.0562 + {0.0352{\mathbb{i}}}} & {{- 0.1507} - {0.2734{\mathbb{i}}}} & {0.0430 - {0.56671{\mathbb{i}}}} \\ {{- 0.3738} - {0.0597{\mathbb{i}}}} & {0.6782 - {0.1616{\mathbb{i}}}} & {0.3675 - {0.0175{\mathbb{i}}}} & {0.1704 - {0.4541{\mathbb{i}}}} \end{matrix}$ ${V\; 4\left( {:{,{:{,34}}}} \right)} = \begin{matrix} {{- 0.0376} + {0.4566{\mathbb{i}}}} & {{- 0.6004} - {0.0993{\mathbb{i}}}} & {0.1028 - {0.0939{\mathbb{i}}}} & {0.6203 + {0.1250{\mathbb{i}}}} \\ {{- 0.1481} + {0.2688{\mathbb{i}}}} & {{- 0.4515} - {0.2985{\mathbb{i}}}} & {{- 0.0297} + {0.3112{\mathbb{i}}}} & {{- 0.5456} - {0.4663{\mathbb{i}}}} \\ {{- 0.1588} + {0.0744{\mathbb{i}}}} & {{- 0.1174} + {0.2648{\mathbb{i}}}} & {{- 0.8756} + {0.2308{\mathbb{i}}}} & {{- 0.0070} + {0.2556{\mathbb{i}}}} \\ {{- 0.5613} + {0.5917{\mathbb{i}}}} & {0.3667 + {0.3441{\mathbb{i}}}} & {0.1195 - {0.2205{\mathbb{i}}}} & {{- 0.0920} - {0.1031{\mathbb{i}}}} \end{matrix}$ ${V\; 4\left( {:{,{:{,35}}}} \right)} = \begin{matrix} {0.2356 + {0.5396{\mathbb{i}}}} & {{- 0.2530} + {0.3209{\mathbb{i}}}} & {{- 0.4814} + {0.3797{\mathbb{i}}}} & {{- 0.1906} - {0.2722{\mathbb{i}}}} \\ {0.6510 - {0.4238{\mathbb{i}}}} & {{- 0.2885} - {0.0485{\mathbb{i}}}} & {{- 0.4053} - {0.3563{\mathbb{i}}}} & {0.1239 + {0.0673{\mathbb{i}}}} \\ {0.1311 + {0.0167{\mathbb{i}}}} & {{- 0.0415} - {0.7379{\mathbb{i}}}} & {{- 0.0707} + {0.5502{\mathbb{i}}}} & {0.3560 - {0.0416{\mathbb{i}}}} \\ {{- 0.0198} - {0.1791{\mathbb{i}}}} & {0.1649 - {0.4171{\mathbb{i}}}} & {{- 0.1172} - {0.1069{\mathbb{i}}}} & {{- 0.7392} - {0.4413{\mathbb{i}}}} \end{matrix}$ ${V\; 4\left( {:{,{:{,36}}}} \right)} = \begin{matrix} {0.4535 + {0.1524{\mathbb{i}}}} & {0.2821 - {0.1279{\mathbb{i}}}} & {{- 0.5649} + {0.3170{\mathbb{i}}}} & {0.3753 + {0.3388{\mathbb{i}}}} \\ {0.1220 + {0.1943{\mathbb{i}}}} & {{- 0.0004} - {0.0348{\mathbb{i}}}} & {0.5468 - {0.3850{\mathbb{i}}}} & {0.6392 + {0.3007{\mathbb{i}}}} \\ {{- 0.7044} + {0.2811{\mathbb{i}}}} & {{- 0.2923} + {0.0831{\mathbb{i}}}} & {{- 0.2695} + {0.2250{\mathbb{i}}}} & {0.4521 - {0.0693{\mathbb{i}}}} \\ {{- 0.0597} + {0.3738{\mathbb{i}}}} & {{- 0.0749} - {0.8971{\mathbb{i}}}} & {{- 0.0004} - {0.0994{\mathbb{i}}}} & {{- 0.1722} - {0.0816{\mathbb{i}}}} \end{matrix}$

${V\; 4\left( {:{,{:{,37}}}} \right)} = \begin{matrix} {{- 0.4472} - {0.1208{\mathbb{i}}}} & {{- 0.4748} + {0.1205{\mathbb{i}}}} & {{- 0.1361} + {0.1678{\mathbb{i}}}} & {0.6410 + {0.2967{\mathbb{i}}}} \\ {0.4936 + {0.0781{\mathbb{i}}}} & {0.2258 + {0.2637{\mathbb{i}}}} & {{- 0.0187} - {0.5564{\mathbb{i}}}} & {0.4803 + {0.2985{\mathbb{i}}}} \\ {0.6133 + {0.0191{\mathbb{i}}}} & {{- 0.2650} + {0.3289{\mathbb{i}}}} & {{- 0.3000} + {0.5816{\mathbb{i}}}} & {{- 0.0870} + {0.0961{\mathbb{i}}}} \\ {{- 0.1738} - {0.3591{\mathbb{i}}}} & {{- 0.2798} + {0.6188{\mathbb{i}}}} & {{- 0.1970} - {0.4199{\mathbb{i}}}} & {{- 0.3841} - {0.1305{\mathbb{i}}}} \end{matrix}$ ${V\; 4\left( {:{,{:{,38}}}} \right)} = \begin{matrix} {0.2356 + {0.5396{\mathbb{i}}}} & {{- 0.0180} - {0.6214{\mathbb{i}}}} & {{- 0.4511} + {0.0275{\mathbb{i}}}} & {0.2056 + {0.1427{\mathbb{i}}}} \\ {0.4238 + {0.6510{\mathbb{i}}}} & {{- 0.0262} + {0.4269{\mathbb{i}}}} & {0.2038 - {0.1591{\mathbb{i}}}} & {{- 0.3709} - {0.0966{\mathbb{i}}}} \\ {{- 0.1311} - {0.1671{\mathbb{i}}}} & {0.1087 + {0.3811{\mathbb{i}}}} & {{- 0.7745} + {0.3487{\mathbb{i}}}} & {{- 0.2829} - {0.1551{\mathbb{i}}}} \\ {{- 0.1791} + {0.0198{\mathbb{i}}}} & {{- 0.5069} + {0.1291{\mathbb{i}}}} & {{- 0.0765} - {0.0408{\mathbb{i}}}} & {{- 0.2182} + {0.7993{\mathbb{i}}}} \end{matrix}$ ${V\; 4\left( {:{,{:{,39}}}} \right)} = \begin{matrix} 0.5000 & {0.0906 - {0.2217{\mathbb{i}}}} & {0.1089 - {0.3913{\mathbb{i}}}} & {0.6994 + {0.1962{\mathbb{i}}}} \\ {0.4619 + {0.1913{\mathbb{i}}}} & {{- 0.3471} + {0.3239{\mathbb{i}}}} & {{- 0.5370} - {0.3087{\mathbb{i}}}} & {{- 0.3246} + {0.1888{\mathbb{i}}}} \\ {0.3536 + {0.3536{\mathbb{i}}}} & {{- 0.2568} + {0.4202{\mathbb{i}}}} & {0.3284 + {0.4249{\mathbb{i}}}} & {0.2167 - {0.4149{\mathbb{i}}}} \\ {0.1913 + {0.4619{\mathbb{i}}}} & {0.6329 - {0.2724{\mathbb{i}}}} & {{- 0.3826} + {0.1289{\mathbb{i}}}} & {{- 0.0824} - {0.3247{\mathbb{i}}}} \end{matrix}$ ${V\; 4\left( {:{,{:{,40}}}} \right)} = \begin{matrix} {{- 0.0628} + {0.5038{\mathbb{i}}}} & {{- 0.2542} - {0.0969{\mathbb{i}}}} & {0.6056 - {0.2040{\mathbb{i}}}} & {{- 0.2829} - {0.4240{\mathbb{i}}}} \\ {0.3646 + {0.2226{\mathbb{i}}}} & {0.1965 + {0.0675{\mathbb{i}}}} & {{- 0.4536} - {0.0227{\mathbb{i}}}} & {0.2347 - {0.7163{\mathbb{i}}}} \\ {0.2517 + {0.1533{\mathbb{i}}}} & {0.8534 + {0.0091{\mathbb{i}}}} & {0.3944 + {0.0214{\mathbb{i}}}} & {0.0241 + {0.1682{\mathbb{i}}}} \\ {{- 0.6562} - {0.2058{\mathbb{i}}}} & {0.3805 + {0.0988{\mathbb{i}}}} & {{- 0.1788} - {0.4443{\mathbb{i}}}} & {{- 0.2841} - {0.2500{\mathbb{i}}}} \end{matrix}$

${V\; 4\left( {:{,{:{,41}}}} \right)} = \begin{matrix} 0.5000 & {{- 0.0310} + {0.3127{\mathbb{i}}}} & {{- 0.4192} + {0.2451{\mathbb{i}}}} & {{- 0.6324} + {0.1246{\mathbb{i}}}} \\ {{- 0.1913} + {0.4619{\mathbb{i}}}} & {{- 0.4944} - {0.1906{\mathbb{i}}}} & {0.1428 + {0.6550{\mathbb{i}}}} & {{- 0.0352} + {0.1364{\mathbb{i}}}} \\ {{- 0.3536} - {0.3536{\mathbb{i}}}} & {0.0394 + {0.1700{\mathbb{i}}}} & {0.4123 + {0.2162{\mathbb{i}}}} & {{- 0.4882} - {0.5143{\mathbb{i}}}} \\ {0.4619 - {0.1913{\mathbb{i}}}} & {{- 0.1337} - {0.7565{\mathbb{i}}}} & {0.2889 - {0.1209{\mathbb{i}}}} & {{- 0.2467} - {0.0313{\mathbb{i}}}} \end{matrix}$ ${V\; 4\left( {:{,{:{,42}}}} \right)} = \begin{matrix} {0.4184 + {0.4842{\mathbb{i}}}} & {0.0769 + {0.4899{\mathbb{i}}}} & {0.3574 - {0.2210{\mathbb{i}}}} & {{- 0.2987} - {0.2809{\mathbb{i}}}} \\ {0.1085 - {0.0629{\mathbb{i}}}} & {0.0623 - {0.0897{\mathbb{i}}}} & {0.7253 - {0.1454{\mathbb{i}}}} & {0.4881 + {0.4323{\mathbb{i}}}} \\ {0.5948 - {0.4538{\mathbb{i}}}} & {{- 0.4402} + {0.1739{\mathbb{i}}}} & {{- 0.0113} + {0.3414{\mathbb{i}}}} & {0.2179 - {0.2281{\mathbb{i}}}} \\ {{- 0.0601} - {0.1068{\mathbb{i}}}} & {{- 0.7025} - {0.1568{\mathbb{i}}}} & {0.1428 - {0.3730{\mathbb{i}}}} & {{- 0.4909} + {0.2576{\mathbb{i}}}} \end{matrix}$ ${V\; 4\left( {:{,{:{,43}}}} \right)} = \begin{matrix} 0.5000 & {{- 0.1844} + {0.1638{\mathbb{i}}}} & {{- 0.6587} + {0.3087{\mathbb{i}}}} & {{- 0.1539} - {0.3692{\mathbb{i}}}} \\ {{- 0.4619} - {0.1913{\mathbb{i}}}} & {0.3389 - {0.3843{\mathbb{i}}}} & {{- 0.2949} + {0.3309{\mathbb{i}}}} & {0.4206 - {0.3378{\mathbb{i}}}} \\ {0.3536 + {0.3536{\mathbb{i}}}} & {0.4409 - {0.5467{\mathbb{i}}}} & {{- 0.1139} + {0.1683{\mathbb{i}}}} & {{- 0.1714} + {0.4313{\mathbb{i}}}} \\ {{- 0.1913} - {0.4619{\mathbb{i}}}} & {{- 0.1168} - {0.4120{\mathbb{i}}}} & {{- 0.2644} - {0.4039{\mathbb{i}}}} & {{- 0.5775} - {0.0090{\mathbb{i}}}} \end{matrix}$ ${V\; 4\left( {:{,{:{,44}}}} \right)} = \begin{matrix} {{- 0.1083} + {0.4189{\mathbb{i}}}} & {{- 0.1461} + {0.1056{\mathbb{i}}}} & {0.2135 + {0.2312{\mathbb{i}}}} & {{- 0.6745} + {0.4757{\mathbb{i}}}} \\ {0.6032 - {0.3824{\mathbb{i}}}} & {0.1171 - {0.2010{\mathbb{i}}}} & {0.0162 + {0.6417{\mathbb{i}}}} & {{- 0.1534} + {0.0178{\mathbb{i}}}} \\ {0.2610 + {0.1225{\mathbb{i}}}} & {{- 0.9157} - {0.2359{\mathbb{i}}}} & {{- 0.0474} - {0.0483{\mathbb{i}}}} & {0.1159 - {0.0681{\mathbb{i}}}} \\ {{- 0.0597} - {0.4649{\mathbb{i}}}} & {{- 0.0670} - {0.1213{\mathbb{i}}}} & {{- 0.6245} - {0.3072{\mathbb{i}}}} & {{- 0.2668} + {0.4534{\mathbb{i}}}} \end{matrix}$

${V\; 4\left( {:{,{:{,45}}}} \right)} = \begin{matrix} 0.5000 & {0.5295 + {0.2225{\mathbb{i}}}} & {{- 0.1373} - {0.3127{\mathbb{i}}}} & {{- 0.4137} + {0.3638{\mathbb{i}}}} \\ {0.1913 - {0.4619{\mathbb{i}}}} & {0.0632 + {0.0938{\mathbb{i}}}} & {0.1078 + {0.6964{\mathbb{i}}}} & {{- 0.4209} - {0.2518{\mathbb{i}}}} \\ {{- 0.3536} - {0.3536{\mathbb{i}}}} & {{- 0.3442} + {0.5791{\mathbb{i}}}} & {{- 0.4500} - {0.0778{\mathbb{i}}}} & {{- 0.1048} + {0.2769{\mathbb{i}}}} \\ {{- 0.4619} + {0.1913{\mathbb{i}}}} & {0.1494 - {0.4256{\mathbb{i}}}} & {{- 0.4187} - {0.0536{\mathbb{i}}}} & {{- 0.5783} - {0.1840{\mathbb{i}}}} \end{matrix}$ ${V\; 4\left( {:{,{:{,46}}}} \right)} = \begin{matrix} {{- 0.0430} - {0.3791{\mathbb{i}}}} & {{- 0.2534} + {0.0743{\mathbb{i}}}} & {{- 0.3840} + {0.2583{\mathbb{i}}}} & {{- 0.2059} - {0.7267{\mathbb{i}}}} \\ {0.5314 - {0.0969{\mathbb{i}}}} & {{- 0.0802} + {0.0767{\mathbb{i}}}} & {0.0963 + {0.7666{\mathbb{i}}}} & {0.2344 + {0.2101{\mathbb{i}}}} \\ {0.5001 - {0.1416{\mathbb{i}}}} & {{- 0.3092} + {0.6079{\mathbb{i}}}} & {{- 0.0022} - {0.4149{\mathbb{i}}}} & {{- 0.2681} + {0.1439{\mathbb{i}}}} \\ {{- 0.3832} - {0.3817{\mathbb{i}}}} & {{- 0.6565} - {0.1479{\mathbb{i}}}} & {{- 0.1295} + {0.0072{\mathbb{i}}}} & {0.1480 + {0.4646{\mathbb{i}}}} \end{matrix}$ ${V\; 4\left( {:{,{:{,47}}}} \right)} = \begin{matrix} 0.5000 & {{- 0.4933} + {0.2493{\mathbb{i}}}} & {{- 0.1093} - {0.6284{\mathbb{i}}}} & {{- 0.1925} - {0.0244{\mathbb{i}}}} \\ {0.1913 + {0.4619{\mathbb{i}}}} & {{- 0.0119} - {0.1919{\mathbb{i}}}} & {{- 0.4573} - {0.0414{\mathbb{i}}}} & {0.6333 + {0.3179{\mathbb{i}}}} \\ {{- 0.3536} + {0.3536{\mathbb{i}}}} & {0.2792 + {0.2603{\mathbb{i}}}} & {{- 0.2294} - {0.3536{\mathbb{i}}}} & {0.0703 - {0.6494{\mathbb{i}}}} \\ {{- 0.4619} - {0.1913{\mathbb{i}}}} & {{- 0.5616} - {0.4432{\mathbb{i}}}} & {{- 0.4520} + {0.0189{\mathbb{i}}}} & {{- 0.1228} - {0.1357{\mathbb{i}}}} \end{matrix}$ ${V\; 4\left( {:{,{:{,48}}}} \right)} = \begin{matrix} {{- 0.0197} + {0.5406{\mathbb{i}}}} & {0.5987 + {0.4276{\mathbb{i}}}} & {0.1110 + {0.0636{\mathbb{i}}}} & {0.3849 + {0.0409{\mathbb{i}}}} \\ {0.1681 + {0.0640{\mathbb{i}}}} & {{- 0.2253} + {0.0427{\mathbb{i}}}} & {0.9463 - {0.0868{\mathbb{i}}}} & {{- 0.0474} + {0.0990{\mathbb{i}}}} \\ {0.2143 - {0.3250{\mathbb{i}}}} & {{- 0.1122} + {0.5394{\mathbb{i}}}} & {0.0270 + {0.1660{\mathbb{i}}}} & {0.0832 - {0.7139{\mathbb{i}}}} \\ {{- 0.2387} - {0.6830{\mathbb{i}}}} & {0.2949 - {0.1252{\mathbb{i}}}} & {0.1828 + {0.1374{\mathbb{i}}}} & {0.5310 + {0.1992{\mathbb{i}}}} \end{matrix}$

${V\; 4\left( {:{,{:{,49}}}} \right)} = \begin{matrix} 0.5000 & {{- 0.1299} - {0.1774{\mathbb{i}}}} & {0.0635 + {0.6372{\mathbb{i}}}} & {{- 0.1454} - {0.5201{\mathbb{i}}}} \\ {{- 0.4619} + {0.1913{\mathbb{i}}}} & {{- 0.7371} - {0.3712{\mathbb{i}}}} & {0.1343 - {0.0486{\mathbb{i}}}} & {0.0889 - {0.2013{\mathbb{i}}}} \\ {0.3536 - {0.3536{\mathbb{i}}}} & {{- 0.1244} - {0.1207{\mathbb{i}}}} & {{- 0.2344} - {0.6779{\mathbb{i}}}} & {0.0211 - {0.4528{\mathbb{i}}}} \\ {{- 0.1913} + {0.4619{\mathbb{i}}}} & {0.4090 - {0.2705{\mathbb{i}}}} & {0.0763 - {0.2220{\mathbb{i}}}} & {{- 0.6105} - {0.2859{\mathbb{i}}}} \end{matrix}$ ${V\; 4\left( {:{,{:{,50}}}} \right)} = \begin{matrix} {0.1819 - {0.0589{\mathbb{i}}}} & {{- 0.3263} - {0.7161{\mathbb{i}}}} & {{- 0.0074} - {0.1928{\mathbb{i}}}} & {{- 0.3735} + {0.4091{\mathbb{i}}}} \\ {0.3368 - {0.1998{\mathbb{i}}}} & {{- 0.0538} + {0.0491{\mathbb{i}}}} & {{- 0.7846} + {0.2480{\mathbb{i}}}} & {{- 0.2825} - {0.2907{\mathbb{i}}}} \\ {0.6565 - {0.0556{\mathbb{i}}}} & {0.2749 - {0.2169{\mathbb{i}}}} & {0.0684 + {0.2639{\mathbb{i}}}} & {0.5764 + {0.1916{\mathbb{i}}}} \\ {{- 0.5241} - {0.3183{\mathbb{i}}}} & {0.4347 - {0.2526{\mathbb{i}}}} & {{- 0.4391} - {0.1363{\mathbb{i}}}} & {0.2853 + {0.2800{\mathbb{i}}}} \end{matrix}$ ${V\; 4\left( {:{,{:{,51}}}} \right)} = \begin{matrix} 0.5000 & {{- 0.3550} + {0.4986{\mathbb{i}}}} & {0.2360 - {0.2295{\mathbb{i}}}} & {0.3696 + {0.3610{\mathbb{i}}}} \\ {{- 0.1913} - {0.4619{\mathbb{i}}}} & {{- 0.5979} - {0.4466{\mathbb{i}}}} & {{- 0.1015} - {0.0709{\mathbb{i}}}} & {0.4087 - {0.1032{\mathbb{i}}}} \\ {{- 0.3536} + {0.3536{\mathbb{i}}}} & {0.2014 + {0.0225{\mathbb{i}}}} & {{- 0.1794} + {0.3659{\mathbb{i}}}} & {0.6567 + {0.3341{\mathbb{i}}}} \\ {0.4619 + {0.1913{\mathbb{i}}}} & {{- 0.1636} - {0.0232{\mathbb{i}}}} & {{- 0.8345} + {0.1176{\mathbb{i}}}} & {{- 0.0448} - {0.1024{\mathbb{i}}}} \end{matrix}$ ${V\; 4\left( {:{,{:{,52}}}} \right)} = \begin{matrix} {0.5307 + {0.2465{\mathbb{i}}}} & {0.4315 + {0.2977{\mathbb{i}}}} & {0.5279 - {0.2633{\mathbb{i}}}} & {{- 0.0096} - {0.1863{\mathbb{i}}}} \\ {0.6357 - {0.3878{\mathbb{i}}}} & {{- 0.3022} + {0.0611{\mathbb{i}}}} & {0.0004 + {0.3129{\mathbb{i}}}} & {0.4461 + {0.2312{\mathbb{i}}}} \\ {0.1539 - {0.1260{\mathbb{i}}}} & {{- 0.4238} - {0.3362{\mathbb{i}}}} & {0.4544 + {0.2155{\mathbb{i}}}} & {{- 0.6006} - {0.2328{\mathbb{i}}}} \\ {{- 0.2499} - {0.0328{\mathbb{i}}}} & {0.3188 - {0.4857{\mathbb{i}}}} & {0.5285 + {0.1480{\mathbb{i}}}} & {0.2559 + {0.4820{\mathbb{i}}}} \end{matrix}$

${V\; 4\left( {:{,{:{,53}}}} \right)} = \begin{matrix} 0.5000 & {0.6135 - {0.5466{\mathbb{i}}}} & {0.1363 + {0.0107{\mathbb{i}}}} & {{- 0.0265} - {0.2357{\mathbb{i}}}} \\ {0.4619 - {0.1913{\mathbb{i}}}} & {{- 0.3336} + {0.1681{\mathbb{i}}}} & {0.3619 + {0.5204{\mathbb{i}}}} & {0.4467 - {0.0956{\mathbb{i}}}} \\ {0.3536 - {0.3536{\mathbb{i}}}} & {0.0923 + {0.1646{\mathbb{i}}}} & {{- 0.3085} - {0.5415{\mathbb{i}}}} & {0.4468 + {0.3555{\mathbb{i}}}} \\ {0.1913 - {0.4619{\mathbb{i}}}} & {0.2389 + {0.3044{\mathbb{i}}}} & {{- 0.1937} + {0.3920{\mathbb{i}}}} & {{- 0.5692} + {0.2917{\mathbb{i}}}} \end{matrix}$ ${V\; 4\left( {:{,{:{,54}}}} \right)} = \begin{matrix} {{- 0.4886} + {0.2995{\mathbb{i}}}} & {{- 0.0643} + {0.2882{\mathbb{i}}}} & {0.7408 - {0.0532{\mathbb{i}}}} & {{- 0.0130} - {0.1806{\mathbb{i}}}} \\ {0.4671 + {0.2039{\mathbb{i}}}} & {0.1360 - {0.4277{\mathbb{i}}}} & {0.4029 + {0.4160{\mathbb{i}}}} & {{- 0.4410} - {0.0947{\mathbb{i}}}} \\ {0.5829 + {0.1869{\mathbb{i}}}} & {{- 0.4682} + {0.3860{\mathbb{i}}}} & {0.0628 + {0.1030{\mathbb{i}}}} & {0.4010 - {0.2857{\mathbb{i}}}} \\ {{- 0.1465} - {0.1250{\mathbb{i}}}} & {{- 0.2986} - {0.5040{\mathbb{i}}}} & {{- 0.0741} - {0.3049{\mathbb{i}}}} & {{- 0.0166} - {0.7218{\mathbb{i}}}} \end{matrix}$

The 1^(st), 4^(th), 7^(th), 10^(th), 13^(th), 16^(th), 19^(th), 22^(nd), 25^(th), 28^(th), 31^(st), 34^(th), 37^(th), 40^(th), 43^(rd), and 46^(th) codewords in each of the 6-bit final rank 1 codebook, the final rank 2 codebook, and the final rank 3 codebook are equal to the 16 codewords included in each of the 4-bit rank codebook, the rank 2 codebook, and the rank 3 codebook, disclosed in the above Table 1.

The 6-bit final rank 1 codebook, the final rank 2 codebook, and the final rank 3 codebook may be obtained using column subsets of the rank 4 codebook. Accordingly, the 6-bit final rank 1 codebook, the final rank 2 codebook, the final rank 3 codebook, and the final rank 4 codebook may be expressed as a function of V4, as given by the following Table 14, and may also be expressed using various types of schemes:

Transmit Trans- Codebook mission Transmission Transmission Transmission Index Rank 1 Rank 2 Rank 3 Rank 4 1 V4(:,2,1) V4(:,[1,2],1) V4(:,[1,2,3],1) V4(:,:,1) 2 V4(:,1,7) V4(:,[1,2],7) V4(:,[2,3,4],7) V4(:,:,2) 3 V4(:,1,8) V4(:,[1,2],8) V4(:,[2,3,4],8) V4(:,:,3) 4 V4(:,3,1) V4(:,[1,3],1) V4(:,[1,2,4],1) V4(:,:,4) 5 V4(:,1,9) V4(:,[1,2],9) V4(:,[2,3,4],9) V4(:,:,5) 6 V4(:,1,10) V4(:,[1,2],10) V4(:,[2,3,4],10) V4(:,:,6) 7 V4(:,4,1) V4(:,[1,4],1) V4(:,[1,3,4],1) V4(:,:,7) 8 V4(:,1,11) V4(:,[1,2],11) V4(:,[2,3,4],11) V4(:,:,8) 9 V4(:,1,12) V4(:,[1,2],12) V4(:,[2,3,4],12) V4(:,:,9) 10 V4(:,2,2) V4(:,[2,3],1) V4(:,[2,3,4],1) V4(:,:,10) 11 V4(:,1,13) V4(:,[1,2],13) V4(:,[2,3,4],13) V4(:,:,11) 12 V4(:,1,14) V4(:,[1,2],14) V4(:,[2,3,4],14) V4(:,:,12) 13 V4(:,3,2) V4(:,[2,4],1) V4(:,[1,2,3],2) V4(:,:,13) 14 V4(:,1,15) V4(:,[1,2],15) V4(:,[2,3,4],15) V4(:,:,14) 15 V4(:,1,16) V4(:,[1,2],16) V4(:,[2,3,4],16) V4(:,:,15) 16 V4(:,4,2) V4(:,[3,4],1) V4(:,[1,2,4],2) V4(:,:,16) 17 V4(:,1,17) V4(:,[1,2],17) V4(:,[2,3,4],17) V4(:,:,17) 18 V4(:,1,18) V4(:,[1,2],18) V4(:,[2,3,4],18) V4(:,:,18) 19 V4(:,1,3) V4(:,[1,3],2) V4(:,[1,3,4],2) V4(:,:,19) 20 V4(:,1,19) V4(:,[1,2],19) V4(:,[2,3,4],19) V4(:,:,20) 21 V4(:,1,20) V4(:,[1,2],20) V4(:,[2,3,4],20) V4(:,:,21) 22 V4(:,1,4) V4(:,[1,4],2) V4(:,[2,3,4],2) V4(:,:,22) 23 V4(:,1,21) V4(:,[1,2],21) V4(:,[2,3,4],21) V4(:,:,23) 24 V4(:,1,22) V4(:,[1,2],22) V4(:,[2,3,4],22) V4(:,:,24) 25 V4(:,1,5) V4(:,[2,3],2) V4(:,[1,2,3],3) V4(:,:,25) 26 V4(:,1,23) V4(:,[1,2],23) V4(:,[2,3,4],23) V4(:,:,26) 27 V4(:,1,24) V4(:,[1,2],24) V4(:,[2,3,4],24) V4(:,:,27) 28 V4(:,2,5) V4(:,[2,4],2) V4(:,[1,3,4],3) V4(:,:,28) 29 V4(:,1,25) V4(:,[1,2],25) V4(:,[2,3,4],25) V4(:,:,29) 30 V4(:,1,26) V4(:,[1,2],26) V4(:,[2,3,4],26) V4(:,:,30) 31 V4(:,3,5) V4(:,[1,3],3) V4(:,[1,2,3],4) V4(:,:,31) 32 V4(:,1,27) V4(:,[1,2],27) V4(:,[2,3,4],27) V4(:,:,32) 32 V4(:,1,28) V4(:,[1,2],28) V4(:,[2,3,4],28) V4(:,:,33) 34 V4(:,4,5) V4(:,[1,4],3) V4(:,[1,3,4],4) V4(:,:,34) 35 V4(:,1,29) V4(:,[1,2],29) V4(:,[2,3,4],29) V4(:,:,35) 36 V4(:,1,30) V4(:,[1,2],30) V4(:,[2,3,4],30) V4(:,:,36) 37 V4(:,1,6) V4(:,[1,3],4) V4(:,[1,2,3],5) V4(:,:,37) 38 V4(:,1,31) V4(:,[1,2],31) V4(:,[2,3,4],31) V4(:,:,38) 39 V4(:,1,32) V4(:,[1,2],32) V4(:,[2,3,4],32) V4(:,:,39) 40 V4(:,2,6) V4(:,[1,4],4) V4(:,[1,3,4],5) V4(:,:,40) 41 V4(:,1,33) V4(:,[1,2],33) V4(:,[2,3,4],33) V4(:,:,41) 42 V4(:,1,34) V4(:,[1,2],34) V4(:,[2,3,4],34) V4(:,:,42) 43 V4(:,3,6) V4(:,[1,3],5) V4(:,[1,2,4],6) V4(:,:,43) 44 V4(:,1,35) V4(:,[1,2],35) V4(:,[2,3,4],35) V4(:,:,44) 45 V4(:,1,36) V4(:,[1,2],36) V4(:,[2,3,4],36) V4(:,:,45) 46 V4(:,4,6) V4(:,[2,4],6) V4(:,[2,3,4],6) V4(:,:,46) 47 V4(:,1,37) V4(:,[1,2],37) V4(:,[2,3,4],37) V4(:,:,47) 48 V4(:,1,38) V4(:,[1,2],38) V4(:,[2,3,4],38) V4(:,:,48) 49 V4(:,1,39) V4(:,[1,2],39) V4(:,[2,3,4],39) V4(:,:,49) 50 V4(:,1,40) V4(:,[1,2],40) V4(:,[2,3,4],40) V4(:,:,50) 51 V4(:,1,41) V4(:,[1,2],41) V4(:,[2,3,4],41) V4(:,:,51) 52 V4(:,1,42) V4(:,[1,2],42) V4(:,[2,3,4],42) V4(:,:,52) 53 V4(:,1,43) V4(:,[1,2],43) V4(:,[2,3,4],43) V4(:,:,53) 54 V4(:,1,44) V4(:,[1,2],44) V4(:,[2,3,4],44) V4(:,:,54) 55 V4(:,1,45) V4(:,[1,2],45) V4(:,[2,3,4],45) 56 V4(:,1,46) V4(:,[1,2],46) V4(:,[2,3,4],46) 57 V4(:,1,47) V4(:,[1,2],47) V4(:,[2,3,4],47) 58 V4(:,1,48) V4(:,[1,2],48) V4(:,[2,3,4],48) 59 V4(:,1,49) V4(:,[1,2],49) V4(:,[2,3,4],49) 60 V4(:,1,50) V4(:,[1,2],50) V4(:,[2,3,4],50) 61 V4(:,1,51) V4(:,[1,2],51) V4(:,[2,3,4],51) 62 V4(:,1,52) V4(:,[1,2],52) V4(:,[2,3,4],52) 63 V4(:,1,53) V4(:,[1,2],53) V4(:,[2,3,4],53) 64 V4(:,1,54) V4(:,[1,2],54) V4(:,[2,3,4],54)

4. Fourth Scheme to Design a 6-Bit Codebook

(1) Operation 1:

The following 4-bit codebook corresponding to the transmission rank 1 may be obtained from the 4-bit codebook disclosed in the above Table 1. Here, the 4-bit codebook corresponding to the transmission rank 1 disclosed in the above Table 1 is referred to as a base codebook. base_cbk(:,1:16)=[C _(1,1) . . . C _(16,1)] with C _(i,1) taken from table 1

(2) Operation 2:

A local codebook local_cbk may be defined, for example, as follows:

${{local\_ cbk} = \begin{bmatrix} \begin{matrix} {0.0975 +} \\ {0.4904{\mathbb{i}}} \end{matrix} & \begin{matrix} {0.0975 +} \\ {0.4904{\mathbb{i}}} \end{matrix} & \begin{matrix} {0.0975 +} \\ {0.4904{\mathbb{i}}} \end{matrix} & \begin{matrix} {0.0975 +} \\ {0.4904{\mathbb{i}}} \end{matrix} \\ \begin{matrix} {{- 0.4904} +} \\ {0.0975{\mathbb{i}}} \end{matrix} & \begin{matrix} {{- 0.0975} -} \\ {0.4904{\mathbb{i}}} \end{matrix} & \begin{matrix} {0.4904 -} \\ {0.0975{\mathbb{i}}} \end{matrix} & \begin{matrix} {0.0975 +} \\ {0.4904{\mathbb{i}}} \end{matrix} \\ \begin{matrix} {{- 0.4904} +} \\ {0.0975{\mathbb{i}}} \end{matrix} & \begin{matrix} {0.4904 -} \\ {0.0975{\mathbb{i}}} \end{matrix} & \begin{matrix} {{- 0.4904} +} \\ {0.0975{\mathbb{i}}} \end{matrix} & \begin{matrix} {0.4904 -} \\ {0.0975{\mathbb{i}}} \end{matrix} \\ \begin{matrix} {0.3536 +} \\ {0.3536{\mathbb{i}}} \end{matrix} & \begin{matrix} {0.3536 -} \\ {0.3536{\mathbb{i}}} \end{matrix} & \begin{matrix} {{- 0.3536} -} \\ {0.3536{\mathbb{i}}} \end{matrix} & \begin{matrix} {{- 0.3536} +} \\ {0.3536{\mathbb{i}}} \end{matrix} \end{bmatrix}};$

(3) Operation 3:

r=abs(local_cbk); phase_local_cbk=local_cbk./r; alpha=0.9835;

A localized codebook localized_cbk may be expressed, for example, as follows:

localized_cbk= [sqrt(1−alpha{circumflex over ( )}2*(1−(r(1,1)){circumflex over ( )}2))*phase_local_cbk(1,1),sqrt(1−alpha{circumflex over ( )}2*(1− (r(1,2)){circumflex over ( )}2))*phase_local_cbk(1,2),sqrt(1−alpha{circumflex over ( )}2*(1−(r(1,3)){circumflex over ( )}2))*phase_local_cbk(1,3),sqrt(1− alpha{circumflex over ( )}2*(1−r(1,4)){circumflex over ( )}2))*phase_local_cbk(1,4);... alpha*r(2,1)*phase_local_cbk(2,1),alpha*r(2,2)*phase_local_cbk(2,2),alpha*r(2,3)*phase_local_cb k(2,3),alpha*r(2,4)*phase_local_cbk(2,4);... alpha*r(3,1)*phase_local_cbk(3,1),alpha*r(3,2)*phase_local_cbk(3,2),alpha*r(3,3)*phase_local_cb k(3,3),alpha*r(3,4)*phase_local_cbk(3,4);... alpha*r(4,1)*phase_local_cbk(4,1),alpha*r(4,2)*phase_local_cbk(4,2),alpha*r(4,3)*phase_local_cb k(4,3),alpha*r(4,4)*phase_local_cbk(4,4)];

Where the localized codebook localized_cbk is calculated through the aforementioned process, vectors of the localized codebook localized_cbk may be normalized, for example, through the following exemplary process:

for k=1:size(localized_cbk,2) localized_cbk(:,k)=localized_cbk(:,k)/norm(localized_cbk(:,k)); end

(4) Operation 4:

A final codebook final_cbk may be obtained by rotating the normalized localized codebook localized_cbk around the base codebook, for example, through the following exemplary process:

for k=1:16 [U,S,V]=svd(base_cbk(:,k)); %, where an SVD is performed with respect to the elements of the base codebook. R=U′*V; % U′*V*base_cbk(:,1,k)=[1;0;0;0]. % , where R denotes a rotation matrix that rotates the normalized localized codebook localized_cbk around the base codebook. rotated_localized=R′*localized_cbk(:,1:3);%, where only first three vectors of localized_cbk are rotated. final_cbk(:,(k−1)*4+1:k*4)=[base_cbk(:,k),rotated_localized];%, where the base codebook is maintained as a centroid and the base codebook is included in a final 6-bit codebook. end;

Six bits of the final rank 1 codebook may be given by final_cbk. A final rank 2 codebook, a final rank 3 codebook, and a final rank 4 codebook may also be obtained based on the final rank 1 codebook. For example, unitary matrices including columns of the final rank 1 codebook, different from the first 16 vectors of the base codebook, may be obtained. A total of 48 unitary matrices may be obtained. A total of 54 matrices may be obtained including W₁ through W₆, and the 48 unitary matrices.

The final rank 2 codebook may be obtained by taking the first two columns from the 48 unitary matrices, and by taking 16 matrices or codewords from the 4-bit rank 2 codebook disclosed in the above Table 1.

The final rank 3 codebook may be obtained by taking the second through the fourth columns from the 48 unitary matrices, and by taking 16 matrices or codewords from the 4-bit rank 3 codebook disclosed in the above Table 1.

Six bits of the final rank 1 codebook, the final rank 2 codebook, the final rank 3 codebook, and the final rank 4 codebook may be expressed as follows:

-   -   Final Rank 1 Codebook:

${V\; 1\left( {:{,{:{,1}}}} \right)} = \begin{matrix} 0.5000 \\ 0.5000 \\ 0.5000 \\ 0.5000 \end{matrix}$ ${V\; 1\left( {:{,{:{,2}}}} \right)} = \begin{matrix} {0.3260 + {0.6774{\mathbb{i}}}} \\ {0.3254 + {0.1709{\mathbb{i}}}} \\ {0.3254 + {0.1709{\mathbb{i}}}} \\ {{- 0.0250} + {0.4051{\mathbb{i}}}} \end{matrix}$ ${V\; 1\left( {:{,{:{,3}}}} \right)} = \begin{matrix} {0.1499 + {0.0{.347}{\mathbb{i}}}} \\ {0.5009 + {0.3071{\mathbb{i}}}} \\ {0.1505 + {0.5412{\mathbb{i}}}} \\ {0.1505 + {0.5412{\mathbb{i}}}} \end{matrix}$ ${V\; 1\left( {:{,{:{,4}}}} \right)} = \begin{matrix} {0.0918 + {0.3270{\mathbb{i}}}} \\ {0.1311 + {0.6387{\mathbb{i}}}} \\ {0.3473 + {0.0541{\mathbb{i}}}} \\ {0.4815 + {0.4045{\mathbb{i}}}} \end{matrix}$

${V\; 1\left( {:{,{:{,5}}}} \right)} = \begin{matrix} 0.5000 \\ {- 0.5000} \\ 0.5000 \\ {- 0.5000} \end{matrix}$ ${V\; 1\left( {:{,{:{,6}}}} \right)} = \begin{matrix} {0.0918 + {0.3270{\mathbb{i}}}} \\ {{- 0.1311} - {0.6387{\mathbb{i}}}} \\ {0.2473 + {0.0541{\mathbb{i}}}} \\ {{- 0.4815} - {0.4045{\mathbb{i}}}} \end{matrix}$ ${V\; 1\left( {:{,{:{,7}}}} \right)} = \begin{matrix} {0.3841 + {0.3851{\mathbb{i}}}} \\ {0.0056 - {0.3076{\mathbb{i}}}} \\ {0.2285 + {0.6580{\mathbb{i}}}} \\ {{- 0.3448} - {0.0735{\mathbb{i}}}} \end{matrix}$ ${V\; 1\left( {:{,{:{,8}}}} \right)} = \begin{matrix} {0.3260 + {0.6774{\mathbb{i}}}} \\ {{- 0.3254} - {0.1709{\mathbb{i}}}} \\ {0.3254 + {0.1709{\mathbb{i}}}} \\ {0.0250 - {0.4051{\mathbb{i}}}} \end{matrix}$

${V\; 1\left( {:{,{:{,9}}}} \right)} = \begin{matrix} {- 0.5000} \\ {- 0.5000} \\ 0.5000 \\ 0.5000 \end{matrix}$ ${V\; 1\left( {:{,{:{,10}}}} \right)} = \begin{matrix} {{- 0.0918} - {0.3270{\mathbb{i}}}} \\ {{- 0.2473} - {0.0541{\mathbb{i}}}} \\ {0.1311 + {0.6387{\mathbb{i}}}} \\ {0.4815 + {0.4045{\mathbb{i}}}} \end{matrix}$ ${V\; 1\left( {:{,{:{,11}}}} \right)} = \begin{matrix} {{- 0.0337} - {0.6193{\mathbb{i}}}} \\ {{- 0.4621} - {0.5019{\mathbb{i}}}} \\ {0.2280 + {0.1515{\mathbb{i}}}} \\ {0.2280 + {0.1515{\mathbb{i}}}} \end{matrix}$ ${V\; 1\left( {:{,{:{,12}}}} \right)} = \begin{matrix} {{- 0.4422} - {0.0928{\mathbb{i}}}} \\ {{- 0.2479} - {0.5606{\mathbb{i}}}} \\ {0.2479 + {0.5606{\mathbb{i}}}} \\ {0.0137 + {0.2102{\mathbb{i}}}} \end{matrix}$

${V\; 1\left( {:{,{:{,13}}}} \right)} = \begin{matrix} {- 0.5000} \\ 0.5000 \\ 0.5000 \\ {- 0.5000} \end{matrix}$ ${V\; 1\left( {:{,{:{,14}}}} \right)} = \begin{matrix} {{- 0.4422} - {0.0928{\mathbb{i}}}} \\ {0.2479 + {0.5606{\mathbb{i}}}} \\ {0.2479 + {0.5606{\mathbb{i}}}} \\ {{- 0.0137} - {0.2102{\mathbb{i}}}} \end{matrix}$ ${V\; 1\left( {:{,{:{,15}}}} \right)} = \begin{matrix} {{- 0.3841} - {0.3851{\mathbb{i}}}} \\ {{- 0.0056} + {0.3076{\mathbb{i}}}} \\ {0.3448 + {0.0735{\mathbb{i}}}} \\ {{- 0.2285} - {0.6580{\mathbb{i}}}} \end{matrix}$ ${V\; 1\left( {:{,{:{,16}}}} \right)} = \begin{matrix} {{- 0.0918} - {0.3270{\mathbb{i}}}} \\ {0.2473 + {0.0541{\mathbb{i}}}} \\ {0.1311 + {0.6387{\mathbb{i}}}} \\ {{- 0.4815} - {0.4045{\mathbb{i}}}} \end{matrix}$

${V\; 1\left( {:{,{:{,17}}}} \right)} = \begin{matrix} \begin{matrix} 0.5000 \\ {0 + {0.5000{\mathbb{i}}}} \\ 0.5000 \end{matrix} \\ {0 + {0.5000{\mathbb{i}}}} \end{matrix}$ ${V\; 1\left( {:{,{:{,18}}}} \right)} = \begin{matrix} \begin{matrix} {0.3841 + {0.3851{\mathbb{i}}}} \\ {{- 0.3076} - {0.0056{\mathbb{i}}}} \\ {0.3448 + {0.0735{\mathbb{i}}}} \end{matrix} \\ {{- 0.6580} + {0.2285{\mathbb{i}}}} \end{matrix}$ ${V\; 1\left( {:{,{:{,19}}}} \right)} = \begin{matrix} \begin{matrix} {0.0918 + {0.3270{\mathbb{i}}}} \\ {{- 0.0541} + {0.2473{\mathbb{i}}}} \\ {0.1311 + {0.6387{\mathbb{i}}}} \end{matrix} \\ {{- 0.4045} + {0.4815{\mathbb{i}}}} \end{matrix}$ ${V\; 1\left( {:{,{:{,20}}}} \right)} = \begin{matrix} \begin{matrix} {0.0337 + {0.6193{\mathbb{i}}}} \\ {{- 0.5019} + {0.4621{\mathbb{i}}}} \\ {0.2280 + {0.1515{\mathbb{i}}}} \end{matrix} \\ {{- 0.1515} + {0.2280{\mathbb{i}}}} \end{matrix}$

${V\; 1\left( {:{,{:{,21}}}} \right)} = \begin{matrix} \begin{matrix} 0.5000 \\ {0 - {0.5000{\mathbb{i}}}} \\ 0.5000 \end{matrix} \\ {0 - {0.5000{\mathbb{i}}}} \end{matrix}$ ${V\; 1\left( {:{,{:{,22}}}} \right)} = \begin{matrix} \begin{matrix} {0.0337 + {0.6193{\mathbb{i}}}} \\ {0.5019 - {0.4621{\mathbb{i}}}} \\ {0.2280 + {0.1515{\mathbb{i}}}} \end{matrix} \\ {0.1515 - {0.2280{\mathbb{i}}}} \end{matrix}$ ${V\; 1\left( {:{,{:{,23}}}} \right)} = \begin{matrix} \begin{matrix} {0.4422 + {0.0928{\mathbb{i}}}} \\ {0.5606 - {0.2479{\mathbb{i}}}} \\ {0.2479 + {0.5606{\mathbb{i}}}} \end{matrix} \\ {0.2102 - {0.0137{\mathbb{i}}}} \end{matrix}$ ${V\; 1\left( {:{,{:{,24}}}} \right)} = \begin{matrix} \begin{matrix} {0.3841 + {0.3851{\mathbb{i}}}} \\ {0.3076 + {0.0056{\mathbb{i}}}} \\ {0.3448 + {0.0735{\mathbb{i}}}} \end{matrix} \\ {0.6580 - {0.2285{\mathbb{i}}}} \end{matrix}$

${V\; 1\left( {:{,{:{,25}}}} \right)} = \begin{matrix} \begin{matrix} {- 0.5000} \\ {0 - {0.5000{\mathbb{i}}}} \\ 0.5000 \end{matrix} \\ {0 + {0.5000{\mathbb{i}}}} \end{matrix}$ ${V\; 1\left( {:{,{:{,26}}}} \right)} = \begin{matrix} \begin{matrix} {{- 0.3841} - {0.3851{\mathbb{i}}}} \\ {0.3076 + {0.0056{\mathbb{i}}}} \\ {0.2285 + {0.6580{\mathbb{i}}}} \end{matrix} \\ {{- 0.0735} + {0.3448{\mathbb{i}}}} \end{matrix}$ ${V\; 1\left( {:{,{:{,27}}}} \right)} = \begin{matrix} \begin{matrix} {{- 0.3260} - {0.6774{\mathbb{i}}}} \\ {0.1709 - {0.3254{\mathbb{i}}}} \\ {0.3254 + {0.1709{\mathbb{i}}}} \end{matrix} \\ {{- 0.4051} - {0.0250{\mathbb{i}}}} \end{matrix}$ ${V\; 1\left( {:{,{:{,28}}}} \right)} = \begin{matrix} \begin{matrix} {{- 0.1499} - {0.0347{\mathbb{i}}}} \\ {0.3017 - {0.5009{\mathbb{i}}}} \\ {0.1505 + {0.5412{\mathbb{i}}}} \end{matrix} \\ {{- 0.5412} + {0.1505{\mathbb{i}}}} \end{matrix}$

${V\; 1\left( {:{,{:{,29}}}} \right)} = \begin{matrix} \begin{matrix} {- 0.5000} \\ {0 + {0.5000{\mathbb{i}}}} \\ 0.5000 \end{matrix} \\ {0 - {0.5000{\mathbb{i}}}} \end{matrix}$ ${V\; 1\left( {:{,{:{,30}}}} \right)} = \begin{matrix} \begin{matrix} {{- 0.1499} - {0.0347{\mathbb{i}}}} \\ {{- 0.3071} + {0.5009{\mathbb{i}}}} \\ {0.1505 + {0.5412{\mathbb{i}}}} \end{matrix} \\ {0.5412 - {0.1505{\mathbb{i}}}} \end{matrix}$ ${V\; 1\left( {:{,{:{,31}}}} \right)} = \begin{matrix} \begin{matrix} {{- 0.0918} - {0.3270{\mathbb{i}}}} \\ {{- 0.6387} + {0.1311{\mathbb{i}}}} \\ {0.2473 + {0.0541{\mathbb{i}}}} \end{matrix} \\ {0.4045 - {0.4815{\mathbb{i}}}} \end{matrix}$ ${V\; 1\left( {:{,{:{,32}}}} \right)} = \begin{matrix} \begin{matrix} {{- 0.3841} - {0.3851{\mathbb{i}}}} \\ {{- 0.3076} - {0.0056{\mathbb{i}}}} \\ {0.2285 + {0.6580{\mathbb{i}}}} \end{matrix} \\ {0.0735 - {0.3448{\mathbb{i}}}} \end{matrix}$

${V\; 1\left( {:{,{:{,33}}}} \right)} = \begin{matrix} \begin{matrix} 0.5000 \\ 0.5000 \\ 0.5000 \end{matrix} \\ {- 0.5000} \end{matrix}$ ${V\; 1\left( {:{,{:{,34}}}} \right)} = \begin{matrix} \begin{matrix} {0.0337 + {0.6193{\mathbb{i}}}} \\ {0.2280 + {0.1515{\mathbb{i}}}} \\ {0.2280 + {0.1515{\mathbb{i}}}} \end{matrix} \\ {{- 0.4621} - {0.5019{\mathbb{i}}}} \end{matrix}$ ${V\; 1\left( {:{,{:{,35}}}} \right)} = \begin{matrix} \begin{matrix} {0.0918 + {0.3270{\mathbb{i}}}} \\ {0.4815 + {0.4045{\mathbb{i}}}} \\ {0.1311 + {0.6387{\mathbb{i}}}} \end{matrix} \\ {{- 0.2473} + {0.0541{\mathbb{i}}}} \end{matrix}$ ${V\; 1\left( {:{,{:{,36}}}} \right)} = \begin{matrix} \begin{matrix} {0.3841 + {0.3851{\mathbb{i}}}} \\ {0.2285 + {0.6580{\mathbb{i}}}} \\ {0.3448 + {0.0735{\mathbb{i}}}} \end{matrix} \\ {0.0056 - {0.3076{\mathbb{i}}}} \end{matrix}$

${V\; 1\left( {:{,{:{,37}}}} \right)} = \begin{matrix} \begin{matrix} 0.5000 \\ {0 + {0.5000{\mathbb{i}}}} \\ {- 0.5000} \end{matrix} \\ {0 + {0.5000{\mathbb{i}}}} \end{matrix}$ ${V\; 1\left( {:{,{:{,38}}}} \right)} = \begin{matrix} \begin{matrix} {0.4422 + {0.0928{\mathbb{i}}}} \\ {{- 0.2102} + {0.0137{\mathbb{i}}}} \\ {{- 0.2479} - {0.5606{\mathbb{i}}}} \end{matrix} \\ {{- 0.5606} + {0.2479{\mathbb{i}}}} \end{matrix}$ ${V\; 1\left( {:{,{:{,39}}}} \right)} = \begin{matrix} \begin{matrix} {0.0337 + {0.6193{\mathbb{i}}}} \\ {{- 0.1515} + {0.2280{\mathbb{i}}}} \\ {{- 0.2280} - {0.1515{\mathbb{i}}}} \end{matrix} \\ {{- 0.5019} + {0.4621{\mathbb{i}}}} \end{matrix}$ ${V\; 1\left( {:{,{:{,40}}}} \right)} = \begin{matrix} \begin{matrix} {0.0918 + {0.3270{\mathbb{i}}}} \\ {{- 0.4045} + {0.4815{\mathbb{i}}}} \\ {{- 0.1311} - {0.6387{\mathbb{i}}}} \end{matrix} \\ {{- 0.0541} + {0.2473{\mathbb{i}}}} \end{matrix}$

${V\; 1\left( {:{,{:{,41}}}} \right)} = \begin{matrix} \begin{matrix} 0.5000 \\ {- 0.5000} \\ 0.5000 \end{matrix} \\ 0.5000 \end{matrix}$ ${V\; 1\left( {:{,{:{,42}}}} \right)} = \begin{matrix} \begin{matrix} {0.3841 + {0.3851{\mathbb{i}}}} \\ {{- 0.2285} - {0.6580{\mathbb{i}}}} \\ {0.3448 + {0.0735{\mathbb{i}}}} \end{matrix} \\ {{- 0.0056} + {0.3076{\mathbb{i}}}} \end{matrix}$ ${V\; 1\left( {:{,{:{,43}}}} \right)} = \begin{matrix} \begin{matrix} {0.4422 + {0.0928{\mathbb{i}}}} \\ {{- 0.0137} - {0.2102{\mathbb{i}}}} \\ {0.2479 + {0.5606{\mathbb{i}}}} \end{matrix} \\ {0.2470 + {0.5606{\mathbb{i}}}} \end{matrix}$ ${V\; 1\left( {:{,{:{,44}}}} \right)} = \begin{matrix} \begin{matrix} {0.0337 + {0.6193{\mathbb{i}}}} \\ {{- 0.2280} - {0.1515{\mathbb{i}}}} \\ {0.2280 + {0.1515{\mathbb{i}}}} \end{matrix} \\ {0.4621 + {0.5019{\mathbb{i}}}} \end{matrix}$

${V\; 1\left( {:{,{:{,45}}}} \right)} = \begin{matrix} \begin{matrix} 0.5000 \\ {0 - {0.5000{\mathbb{i}}}} \\ {- 0.5000} \end{matrix} \\ {0 - {0.5000{\mathbb{i}}}} \end{matrix}$ ${V\; 1\left( {:{,{:{,46}}}} \right)} = \begin{matrix} \begin{matrix} {0.0918 + {0.3270{\mathbb{i}}}} \\ {0.4045 - {0.4815{\mathbb{i}}}} \\ {{- 0.1311} - {0.6387{\mathbb{i}}}} \end{matrix} \\ {0.0541 - {0.2473{\mathbb{i}}}} \end{matrix}$ ${V\; 1\left( {:{,{:{,47}}}} \right)} = \begin{matrix} \begin{matrix} {0.3841 + {0.3851{\mathbb{i}}}} \\ {0.6580 - {0.2285{\mathbb{i}}}} \\ {{- 0.3448} - {0.0735{\mathbb{i}}}} \end{matrix} \\ {0.3076 + {0.0056{\mathbb{i}}}} \end{matrix}$ ${V\; 1\left( {:{,{:{,48}}}} \right)} = \begin{matrix} \begin{matrix} {0.4422 + {0.0928{\mathbb{i}}}} \\ {0.2102 - {0.0137{\mathbb{i}}}} \\ {{- 0.2479} - {0.5606{\mathbb{i}}}} \end{matrix} \\ {0.5606 - {0.2479{\mathbb{i}}}} \end{matrix}$

${V\; 1\left( {:{,{:{,49}}}} \right)} = \begin{matrix} \begin{matrix} 0.5000 \\ {0.3536 + {0.3536{\mathbb{i}}}} \\ {0 + {0.5000{\mathbb{i}}}} \end{matrix} \\ {{- 0.3536} + {0.3536{\mathbb{i}}}} \end{matrix}$ ${V\; 1\left( {:{,{:{,50}}}} \right)} = \begin{matrix} \begin{matrix} {0.3841 + {0.3851{\mathbb{i}}}} \\ {0.0022 + {0.1690{\mathbb{i}}}} \\ {{- 0.3076} - {0.0056{\mathbb{i}}}} \end{matrix} \\ {{- 0.7536} - {0.1140{\mathbb{i}}}} \end{matrix}$ ${V\; 1\left( {:{,{:{,51}}}} \right)} = \begin{matrix} \begin{matrix} {{- 0.1560} + {0.4926{\mathbb{i}}}} \\ {0.0837 + {0.4175{\mathbb{i}}}} \\ {{- 0.4597} + {0.3989{\mathbb{i}}}} \end{matrix} \\ {{- 0.4175} + {0.0837{\mathbb{i}}}} \end{matrix}$ ${V\; 1\left( {:{,{:{,52}}}} \right)} = \begin{matrix} \begin{matrix} {0.3841 + {0.3851{\mathbb{i}}}} \\ {{- 0.1140} + {0.7536{\mathbb{i}}}} \\ {{- 0.3076} - {0.0056{\mathbb{i}}}} \end{matrix} \\ {{- 0.1690} + {0.0022{\mathbb{i}}}} \end{matrix}$

${V\; 1\left( {:{,{:{,53}}}} \right)} = \begin{matrix} \begin{matrix} 0.5000 \\ {{- 0.3536} + {0.3536{\mathbb{i}}}} \\ {0 - {0.5000{\mathbb{i}}}} \end{matrix} \\ {0.3536 + {0.3536{\mathbb{i}}}} \end{matrix}$ ${V\; 1\left( {:{,{:{,54}}}} \right)} = \begin{matrix} \begin{matrix} {0.3396 + {0.1614{\mathbb{i}}}} \\ {{- 0.3400} - {0.3060{\mathbb{i}}}} \\ {0.3493 - {0.5641{\mathbb{i}}}} \end{matrix} \\ {{- 0.3060} + {0.3400{\mathbb{i}}}} \end{matrix}$ ${V\; 1\left( {:{,{:{,55}}}} \right)} = \begin{matrix} \begin{matrix} {0.3841 + {0.3851{\mathbb{i}}}} \\ {{- 0.1690} + {0.0022{\mathbb{i}}}} \\ {0.3076 + {0.0056{\mathbb{i}}}} \end{matrix} \\ {{- 0.1140} + {0.7536{\mathbb{i}}}} \end{matrix}$ ${V\; 1\left( {:{,{:{,56}}}} \right)} = \begin{matrix} \begin{matrix} {{- 0.1560} + {0.4926{\mathbb{i}}}} \\ {{- 0.4175} + {0.0837{\mathbb{i}}}} \\ {0.4597 - {0.3989{\mathbb{i}}}} \end{matrix} \\ {0.0837 + {0.4175{\mathbb{i}}}} \end{matrix}$

${V\; 1\left( {:{,{:{,57}}}} \right)} = \begin{matrix} \begin{matrix} 0.5000 \\ {{- 0.3536} - {0.3536{\mathbb{i}}}} \\ {0 + {0.5000{\mathbb{i}}}} \end{matrix} \\ {0.3536 - {0.3536{\mathbb{i}}}} \end{matrix}$ ${V\; 1\left( {:{,{:{,58}}}} \right)} = \begin{matrix} \begin{matrix} {0.3841 + {0.3851{\mathbb{i}}}} \\ {0.1140 - {0.7536{\mathbb{i}}}} \\ {{- 0.3076} - {0.0056{\mathbb{i}}}} \end{matrix} \\ {0.1690 - {0.0022{\mathbb{i}}}} \end{matrix}$ ${V\; 1\left( {:{,{:{,59}}}} \right)} = \begin{matrix} \begin{matrix} {0.3396 + {0.1614{\mathbb{i}}}} \\ {0.3060 - {0.3400{\mathbb{i}}}} \\ {{- 0.3493} + {0.5641{\mathbb{i}}}} \end{matrix} \\ {0.3400 + {0.3060{\mathbb{i}}}} \end{matrix}$ ${V\; 1\left( {:{,{:{,60}}}} \right)} = \begin{matrix} \begin{matrix} {0.3841 + {0.3851{\mathbb{i}}}} \\ {{- 0.0022} - {0.1690{\mathbb{i}}}} \\ {{- 0.3076} - {0.0056{\mathbb{i}}}} \end{matrix} \\ {0.7536 + {0.1140{\mathbb{i}}}} \end{matrix}$

${V\; 1\left( {:{,{:{,61}}}} \right)} = \begin{matrix} 0.5000 \\ {0.3536 - {0.3536{\mathbb{i}}}} \\ {0. - {0.5000{\mathbb{i}}}} \\ {{- 0.3536} - {0.3536{\mathbb{i}}}} \end{matrix}$ ${V\; 1\left( {:{,{:{,62}}}} \right)} = \begin{matrix} {{- 0.1560} + {0.4926{\mathbb{i}}}} \\ {0.4175 - {0.0837{\mathbb{i}}}} \\ {0.4597 - {0.3989{\mathbb{i}}}} \\ {{- 0.0837} - {0.4175{\mathbb{i}}}} \end{matrix}$ ${V\; 1\left( {:{,{:{,63}}}} \right)} = \begin{matrix} {0.3841 + {0.3851{\mathbb{i}}}} \\ {0.7536 + {0.1140{\mathbb{i}}}} \\ {0.3076 + {0.0056{\mathbb{i}}}} \\ {{- 0.0022} - {0.1690{\mathbb{i}}}} \end{matrix}$ ${V\; 1\left( {:{,{:{,64}}}} \right)} = \begin{matrix} {0.3396 + {0.1614{\mathbb{i}}}} \\ {0.3400 + {0.3060{\mathbb{i}}}} \\ {0.3493 - {0.5641{\mathbb{i}}}} \\ {0.3060 - {0.3400{\mathbb{i}}}} \end{matrix}$

-   -   Final Rank 2 Codebook:

${V\; 2\left( {:{,{:{,1}}}} \right)} = \begin{matrix} 0.5000 & 0.5000 \\ 0.5000 & {- 0.5000} \\ 0.5000 & 0.5000 \\ 0.5000 & {- 0.5000} \end{matrix}$ ${V\; 2\left( {:{,{:{,2}}}} \right)} = \begin{matrix} {0.3260 + {0.6774{\mathbb{i}}}} & {0.4688 + {0.2170{\mathbb{i}}}} \\ {0.3254 + {0.1709{\mathbb{i}}}} & {{- 0.2948} - {0.0700{\mathbb{i}}}} \\ {0.3254 + {0.1709{\mathbb{i}}}} & {{- 0.4598} - {0.2965{\mathbb{i}}}} \\ {{- 0.0250} + {0.4051{\mathbb{i}}}} & {{- 0.5846} - {0.0154{\mathbb{i}}}} \end{matrix}$ ${V\; 2\left( {:{,{:{,3}}}} \right)} = \begin{matrix} {0.1499 + {0.0347{\mathbb{i}}}} & {{- 0.4377} + {0.4498{\mathbb{i}}}} \\ {0.5009 + {0.3071{\mathbb{i}}}} & {{- 0.1173} + {0.2079{\mathbb{i}}}} \\ {0.1505 + {0.5412{\mathbb{i}}}} & {{- 0.2098} + {0.2390{\mathbb{i}}}} \\ {0.1505 + {0.5412{\mathbb{i}}}} & {0.6133 - {0.26821{\mathbb{i}}}} \end{matrix}$ ${V\; 2\left( {:{,{:{,4}}}} \right)} = \begin{matrix} {0.0918 + {0.3270{\mathbb{i}}}} & {0.4585 - {0.3521{\mathbb{i}}}} \\ {0.1311 + {0.6387{\mathbb{i}}}} & {{- 0.5369} + {0.2492{\mathbb{i}}}} \\ {0.2473 + {0.0541{\mathbb{i}}}} & {{- 0.1391} + {0.3154{\mathbb{i}}}} \\ {0.4815 + {0.4045{\mathbb{i}}}} & {0.2872 - {0.3378{\mathbb{i}}}} \end{matrix}$

${V\; 2\left( {:{,{:{,5}}}} \right)} = \begin{matrix} 0.5000 & {- 0.5000} \\ 0.5000 & {- 0.5000} \\ 0.5000 & 0.5000 \\ 0.5000 & 0.5000 \end{matrix}$ ${V\; 2\left( {:{,{:{,6}}}} \right)} = \begin{matrix} {0.0918 + {0.3270{\mathbb{i}}}} & {{- 0.7326} - {0.3303{\mathbb{i}}}} \\ {{- 0.1311} - {0.6387{\mathbb{i}}}} & {{- 0.1521} + {0.1245{\mathbb{i}}}} \\ {0.2473 + {0.0541{\mathbb{i}}}} & {0.0048 - {0.3893{\mathbb{i}}}} \\ {{- 0.4815} - {0.4045{\mathbb{i}}}} & {{- 0.3093} - {0.2615{\mathbb{i}}}} \end{matrix}$ ${V\; 2\left( {:{,{:{,7}}}} \right)} = \begin{matrix} {0.3841 + {0.3851{\mathbb{i}}}} & {{- 0.4259} - {06865{\mathbb{i}}}} \\ {0.0056 - {0.3076{\mathbb{i}}}} & {0.0000 - {0.0163{\mathbb{i}}}} \\ {0.2285 + {0.6580{\mathbb{i}}}} & {{- 0.0029} + {0.5625{\mathbb{i}}}} \\ {{- 0.3448} - {0.0735{\mathbb{i}}}} & {{- 0.1664} + {0.0538{\mathbb{i}}}} \end{matrix}$ ${V\; 2\left( {:{,{:{,8}}}} \right)} = \begin{matrix} {0.3260 + {0.6774{\mathbb{i}}}} & {{- 0.2522} + {0.2830{\mathbb{i}}}} \\ {{- 0.3254} - {0.1709{\mathbb{i}}}} & {{- 0.2511} + {0.3562{\mathbb{i}}}} \\ {0.3254 + {0.1709{\mathbb{i}}}} & {{- 0.6354} - {0.2810{\mathbb{i}}}} \\ {0.0250 - {0.4051{\mathbb{i}}}} & {{- 0.2798} - {0.3246{\mathbb{i}}}} \end{matrix}$

${V\; 2\left( {:{,{:{,9}}}} \right)} = \begin{matrix} 0.5000 & {- 0.5000} \\ 0.5000 & 0.5000 \\ 0.5000 & 0.5000 \\ 0.5000 & {- 0.5000} \end{matrix}$ ${V\; 2\left( {:{,{:{,10}}}} \right)} = \begin{matrix} {{- 0.0918} - {0.3270{\mathbb{i}}}} & {0.4517 - {0.1172{\mathbb{i}}}} \\ {{- 0.2473} - {0.0541{\mathbb{i}}}} & {{- 0.3568} + {0.3992{\mathbb{i}}}} \\ {0.1311 + {0.6387{\mathbb{i}}}} & {0.0918 + {0.4242{\mathbb{i}}}} \\ {0.4815 + {0.4045{\mathbb{i}}}} & {{- 0.3837} - {0.3999{\mathbb{i}}}} \end{matrix}$ ${V\; 2\left( {:{,{:{,11}}}} \right)} = \begin{matrix} {{- 0.0337} - {0.6193{\mathbb{i}}}} & {0.3983 + {0.5597{\mathbb{i}}}} \\ {{- 0.4621} - {0.5019{\mathbb{i}}}} & {{- 0.5583} + {0.0724{\mathbb{i}}}} \\ {0.2280 + {0.1515{\mathbb{i}}}} & {0.2543 + {0.2747{\mathbb{i}}}} \\ {0.2280 + {0.1515{\mathbb{i}}}} & {{- 0.0069} + {0.2664{\mathbb{i}}}} \end{matrix}$ ${V\; 2\left( {:{,{:{,12}}}} \right)} = \begin{matrix} {{- 0.4422} - {0.0928{\mathbb{i}}}} & {0.3609 + {0.0743{\mathbb{i}}}} \\ {{- 0.2479} - {0.5606{\mathbb{i}}}} & {{- 0.0447} - {0.2624{\mathbb{i}}}} \\ {0.2479 + {0.5606{\mathbb{i}}}} & {0.2646 - {02157{\mathbb{i}}}} \\ {0.0137 + {0.2102{\mathbb{i}}}} & {{- 0.7438} + {0.3516{\mathbb{i}}}} \end{matrix}$

${V\; 2\left( {:{,{:{,13}}}} \right)} = \begin{matrix} 0.5000 & {- 0.5000} \\ {- 0.5000} & {- 0.5000} \\ 0.5000 & 0.5000 \\ {- 0.5000} & 0.5000 \end{matrix}$ ${V\; 2\left( {:{,{:{,14}}}} \right)} = \begin{matrix} {0.4422 - {0.0928{\mathbb{i}}}} & {{- 0.4197} + {0.4817{\mathbb{i}}}} \\ {0.2479 + {0.5606{\mathbb{i}}}} & {{- 0.2858} - {0.0562{\mathbb{i}}}} \\ {0.2479 + {0.5606{\mathbb{i}}}} & {{- 0.0712} + {0.2154{\mathbb{i}}}} \\ {{- 0.0137} - {0.2102{\mathbb{i}}}} & {0.1024 + {0.6671{\mathbb{i}}}} \end{matrix}$ ${V\; 2\left( {:{,{:{,15}}}} \right)} = \begin{matrix} {{- 0.3841} - {0.3851{\mathbb{i}}}} & {0.3849 + {0.1986{\mathbb{i}}}} \\ {{- 0.0056} + {0.3076{\mathbb{i}}}} & {{- 0.1824} - {0.0094{\mathbb{i}}}} \\ {0.3448 + {0.0735{\mathbb{i}}}} & {0.2254 + {0.6795{\mathbb{i}}}} \\ {{- 0.2285} - {0.6580{\mathbb{i}}}} & {{- 0.5154} + {0.0292{\mathbb{i}}}} \end{matrix}$ ${V\; 2\left( {:{,{:{,16}}}} \right)} = \begin{matrix} {{- 0.0918} - {0.3270{\mathbb{i}}}} & {0.6638 - {0.4917{\mathbb{i}}}} \\ {0.2473 + {0.0541{\mathbb{i}}}} & {{- 0.1566} - {0.1262{\mathbb{i}}}} \\ {0.1311 + {0.6387{\mathbb{i}}}} & {{- 0.0029} - {0.3366{\mathbb{i}}}} \\ {{- 0.4815} - {0.4045{\mathbb{i}}}} & {{- 0.3978} + {0.0753{\mathbb{i}}}} \end{matrix}$

${V\; 2\left( {:{,{:{,17}}}} \right)} = \begin{matrix} 0.5000 & {- 0.5000} \\ {- 0.5000} & 0.5000 \\ 0.5000 & 0.5000 \\ {- 0.5000} & {- 0.5000} \end{matrix}$ ${V\; 2\left( {:{,{:{,18}}}} \right)} = \begin{matrix} {0.3841 + {0.3851{\mathbb{i}}}} & {0.2420 - {0.2055{\mathbb{i}}}} \\ {{- 0.3076} - {0.0056{\mathbb{i}}}} & {{- 0.5655} - {0.6754{\mathbb{i}}}} \\ {0.3448 + {0.0735{\mathbb{i}}}} & {0.0981 + {0.0467{\mathbb{i}}}} \\ {{- 0.6580} + {0.2285{\mathbb{i}}}} & {0.3298 - {0.0517{\mathbb{i}}}} \end{matrix}$ ${V\; 2\left( {:{,{:{,19}}}} \right)} = \begin{matrix} {0.0918 + {0.3270{\mathbb{i}}}} & {{- 0.2841} - {0.7969{\mathbb{i}}}} \\ {{- 0.0541} + {0.2473{\mathbb{i}}}} & {0.0227 + {0.2058{\mathbb{i}}}} \\ {0.1311 + {0.6387{\mathbb{i}}}} & {0.2505 + {0.3816{\mathbb{i}}}} \\ {{- 0.4045} + {0.4815{\mathbb{i}}}} & {{- 0.0898} - {0.1576{\mathbb{i}}}} \end{matrix}$ ${V\; 2\left( {:{,{:{,20}}}} \right)} = \begin{matrix} {0.0337 + {0.6193{\mathbb{i}}}} & {0.3344 - {0.1640{\mathbb{i}}}} \\ {{- 0.5019} + {0.4621{\mathbb{i}}}} & {{- 0.1475} + {0.2641{\mathbb{i}}}} \\ {0.2280 + {0.1515{\mathbb{i}}}} & {{- 0.1383} + {0.3182{\mathbb{i}}}} \\ {{- 0.1515} + {0.2280{\mathbb{i}}}} & {{- 0.3105} - {0.7437{\mathbb{i}}}} \end{matrix}$

${V\; 2\left( {:{,{:{,21}}}} \right)} = \begin{matrix} {- 0.5000} & {- 0.5000} \\ {- 0.5000} & 0.5000 \\ 0.5000 & 0.5000 \\ 0.5000 & {- 0.5000} \end{matrix}$ ${V\; 2\left( {:{,{:{,22}}}} \right)} = \begin{matrix} {0.0337 + {0.6193{\mathbb{i}}}} & {0.4332 + {0.1612{\mathbb{i}}}} \\ {0.5019 - {0.4621{\mathbb{i}}}} & {{- 0.2499} + {0.4647{\mathbb{i}}}} \\ {0.2280 + {0.1515{\mathbb{i}}}} & {0.1294 + {0.5446{\mathbb{i}}}} \\ {0.1515 - {0.2280{\mathbb{i}}}} & {0.3535 - {0.2640{\mathbb{i}}}} \end{matrix}$ ${V\; 2\left( {:{,{:{,23}}}} \right)} = \begin{matrix} {0.4422 + {0.0928{\mathbb{i}}}} & {0.5948 - {0.2086{\mathbb{i}}}} \\ {0.5606 - {0.2479{\mathbb{i}}}} & {{- 0.6101} + {0.1811{\mathbb{i}}}} \\ {0.2479 + {0.5606{\mathbb{i}}}} & {{- 0.1244} + {0.3625{\mathbb{i}}}} \\ {0.2102 - {0.0137{\mathbb{i}}}} & {{- 0.1261} + {0.1870{\mathbb{i}}}} \end{matrix}$ ${V\; 2\left( {:{,{:{,24}}}} \right)} = \begin{matrix} {0.3841 + {0.3851{\mathbb{i}}}} & {{- 0.7276} + {0.3231{\mathbb{i}}}} \\ {0.3076 + {0.0056{\mathbb{i}}}} & {0.0623 - {0.3109{\mathbb{i}}}} \\ {0.3448 + {0.0735{\mathbb{i}}}} & {{- 0.0316} - {0.3607{\mathbb{i}}}} \\ {0.6580 - {0.2285{\mathbb{i}}}} & {0.1496 - {0.3350{\mathbb{i}}}} \end{matrix}$

${V\; 2\left( {:{,{:{,25}}}} \right)} = \begin{matrix} 0.5000 & {- 0.5000} \\ {0 + {0.5000{\mathbb{i}}}} & {0 - {0.5000{\mathbb{i}}}} \\ 0.5000 & 0.5000 \\ {0 + {0.5000{\mathbb{i}}}} & {0 + {0.5000{\mathbb{i}}}} \end{matrix}$ ${V\; 2\left( {:{,{:{,26}}}} \right)} = \begin{matrix} {{- 0.3841} - {0.3851{\mathbb{i}}}} & {0.2650 + {0.1831{\mathbb{i}}}} \\ {0.3076 + {0.0056{\mathbb{i}}}} & {0.0741 + {0.1142{\mathbb{i}}}} \\ {0.2285 + {0.5680{\mathbb{i}}}} & {0.3746 + {0.4660{\mathbb{i}}}} \\ {{- 0.0735} + {0.3448{\mathbb{i}}}} & {{- 0.0601} - {0.7188{\mathbb{i}}}} \end{matrix}$ ${V\; 2\left( {:{,{:{,27}}}} \right)} = \begin{matrix} {{- 0.3260} - {0.6774{\mathbb{i}}}} & {0.4170 + {0.2673{\mathbb{i}}}} \\ {0.1709 - {0.3254{\mathbb{i}}}} & {{- 0.0605} + {0.0504{\mathbb{i}}}} \\ {0.3254 + {0.1709{\mathbb{i}}}} & {0.3145 + {0.3209{\mathbb{i}}}} \\ {{- 0.4051} - {0.0250{\mathbb{i}}}} & {{- 0.4946} + {0.5495{\mathbb{i}}}} \end{matrix}$ ${V\; 2\left( {:{,{:{,28}}}} \right)} = \begin{matrix} {{- 0.1499} - {0.0347{\mathbb{i}}}} & {{- 0.1568} + {0.5108{\mathbb{i}}}} \\ {0.3071 - {0.5009{\mathbb{i}}}} & {{- 0.2476} + {0.2774{\mathbb{i}}}} \\ {0.1505 + {0.5412{\mathbb{i}}}} & {0.1736 - {0.2279{\mathbb{i}}}} \\ {{- 0.5412} + {0.1505{\mathbb{i}}}} & {{- 0.6443} - {0.2811{\mathbb{i}}}} \end{matrix}$

${V\; 2\left( {:{,{:{,29}}}} \right)} = \begin{matrix} 0.5000 & {- 0.5000} \\ {0 + {0.5000{\mathbb{i}}}} & {0 + {0.5000{\mathbb{i}}}} \\ 0.5000 & 0.5000 \\ {0 + {0.5000{\mathbb{i}}}} & {0 - {0.5000{\mathbb{i}}}} \end{matrix}$ ${V\; 2\left( {:{,{:{,30}}}} \right)} = \begin{matrix} {{- 0.1499} - {0.0347{\mathbb{i}}}} & {{- 0.4915} + {0.5944{\mathbb{i}}}} \\ {{- 0.3071} + {0.5009{\mathbb{i}}}} & {{- 0.1046} - {0.3964{\mathbb{i}}}} \\ {0.1505 + {0.5412{\mathbb{i}}}} & {0.2288 + {0.3710{\mathbb{i}}}} \\ {0.5412 - {0.1505{\mathbb{i}}}} & {{- 0.2089} + {0.0583{\mathbb{i}}}} \end{matrix}$ ${V\; 2\left( {:{,{:{,31}}}} \right)} = \begin{matrix} {{- 0.0918} - {0.3270{\mathbb{i}}}} & {0.0849 - {0.1823{\mathbb{i}}}} \\ {{- 0.6387} + {0.1311{\mathbb{i}}}} & {{- 0.1210} - {0.0233{\mathbb{i}}}} \\ {0.2473 + {0.0541{\mathbb{i}}}} & {0.4127 - {0.7969{\mathbb{i}}}} \\ {0.4045 - {0.4815{\mathbb{i}}}} & {{- 0.0136} + {0.3727{\mathbb{i}}}} \end{matrix}$ ${V\; 2\left( {:{,{:{,32}}}} \right)} = \begin{matrix} {{- 0.3841} - {0.3851{\mathbb{i}}}} & {0.4828 - {0.2545{\mathbb{i}}}} \\ {{- 0.3076} - {0.0056{\mathbb{i}}}} & {{- 0.1939} + {0.1069{\mathbb{i}}}} \\ {0.2285 + {0.6580{\mathbb{i}}}} & {0.3116 + {0.2702{\mathbb{i}}}} \\ {0.0735 - {0.3448{\mathbb{i}}}} & {{- 0.4258} + {0.5492{\mathbb{i}}}} \end{matrix}$

${V\; 2\left( {:{,{:{,33}}}} \right)} = \begin{matrix} 0.5000 & {- 0.5000} \\ {0 - {0.5000{\mathbb{i}}}} & {0 - {0.5000{\mathbb{i}}}} \\ 0.5000 & 0.5000 \\ {0 - {0.5000{\mathbb{i}}}} & {0 + {0.5000{\mathbb{i}}}} \end{matrix}$ ${V\; 2\left( {:{,{:{,34}}}} \right)} = \begin{matrix} {0.0337 + {0.6193{\mathbb{i}}}} & {{- 0.3283} - {0.4339{\mathbb{i}}}} \\ {0.2280 + {0.1515{\mathbb{i}}}} & {{- 0.3886} - {0.2987{\mathbb{i}}}} \\ {0.2280 + {0.1515{\mathbb{i}}}} & {0.2457 - {0.2337{\mathbb{i}}}} \\ {{- 0.4621} - {0.5019{\mathbb{i}}}} & {{- 0.4861} - {0.3354{\mathbb{i}}}} \end{matrix}$ ${V\; 2\left( {:{,{:{,35}}}} \right)} = \begin{matrix} {0.0918 + {0.3270{\mathbb{i}}}} & {{- 0.0348} + {0.4484{\mathbb{i}}}} \\ {0.4815 + {0.4045{\mathbb{i}}}} & {{- 0.4400} + {0.2694{\mathbb{i}}}} \\ {0.1311 + {0.6387{\mathbb{i}}}} & {0.3429 - {0.0049{\mathbb{i}}}} \\ {{- 0.2473} - {0.0541{\mathbb{i}}}} & {0.1991 + {0.06118{\mathbb{i}}}} \end{matrix}$ ${V\; 2\left( {:{,{:{,36}}}} \right)} = \begin{matrix} {0.3841 + {0.3851{\mathbb{i}}}} & {{- 0.2360} - {0.2973{\mathbb{i}}}} \\ {0.2285 + {0.6580{\mathbb{i}}}} & {0.0315 - {0.2366{\mathbb{i}}}} \\ {0.3448 + {0.0735{\mathbb{i}}}} & {0.7285 + {0.0791{\mathbb{i}}}} \\ {0.0056 - {0.3076{\mathbb{i}}}} & {0.4097 - {0.3067{\mathbb{i}}}} \end{matrix}$

${V\; 2\left( {:{,{:{,37}}}} \right)} = \begin{matrix} 0.5000 & {- 0.5000} \\ {0 - {0.5000{\mathbb{i}}}} & {0 + {0.5000{\mathbb{i}}}} \\ 0.5000 & 0.5000 \\ {0 - {0.5000{\mathbb{i}}}} & {0 - {0.5000{\mathbb{i}}}} \end{matrix}$ ${V\; 2\left( {:{,{:{,38}}}} \right)} = \begin{matrix} {0.4422 + {0.0928{\mathbb{i}}}} & {{- 0.5245} - {0.3774{\mathbb{i}}}} \\ {{- 0.2102} + {0.0137{\mathbb{i}}}} & {{- 0.0522} + {0.4805{\mathbb{i}}}} \\ {{- 0.2479} - {0.5606{\mathbb{i}}}} & {0.1023 + {0.0593{\mathbb{i}}}} \\ {{- 0.5606} + {0.2479{\mathbb{i}}}} & {{- 0.5756} - {0.0591{\mathbb{i}}}} \end{matrix}$ ${V\; 2\left( {:{,{:{,39}}}} \right)} = \begin{matrix} {0.0337 + {0.6193{\mathbb{i}}}} & {0.1953 - {0.3057{\mathbb{i}}}} \\ {{- 0.1515} + {0.2280{\mathbb{i}}}} & {{- 0.2685} - {0.1895{\mathbb{i}}}} \\ {{- 0.2280} - {0.1515{\mathbb{i}}}} & {{- 0.7779} - {0.3721{\mathbb{i}}}} \\ {{- 0.5019} + {0.4621{\mathbb{i}}}} & {{- 0.0216} - {0.1283{\mathbb{i}}}} \end{matrix}$ ${V\; 2\left( {:{,{:{,40}}}} \right)} = \begin{matrix} {0.0918 + {0.3270{\mathbb{i}}}} & {{- 0.6035} - {0.3342{\mathbb{i}}}} \\ {{- 0.4045} + {0.4815{\mathbb{i}}}} & {{- 0.1428} + {0.4838{\mathbb{i}}}} \\ {{- 0.1311} - {0.6387{\mathbb{i}}}} & {{- 0.1927} + {0.1626{\mathbb{i}}}} \\ {{- 0.0541} + {0.2473{\mathbb{i}}}} & {{- 0.3638} - {0.2714{\mathbb{i}}}} \end{matrix}$

${V\; 2\left( {:{,{:{,41}}}} \right)} = \begin{matrix} 0.5000 & {- 0.5000} \\ 0.5000 & {0 - {0.5000{\mathbb{i}}}} \\ 0.5000 & 0.5000 \\ 0.5000 & {0 + {0.5000{\mathbb{i}}}} \end{matrix}$ ${V\; 2\left( {:{,{:{,42}}}} \right)} = \begin{matrix} {0.3841 + {0.3851{\mathbb{i}}}} & {0.1371 + {0.7838{\mathbb{i}}}} \\ {{- 0.2285} - {0.6580{\mathbb{i}}}} & {{- 0.0956} + {0.3657{\mathbb{i}}}} \\ {0.3448 + {0.0735{\mathbb{i}}}} & {{- 0.1099} - {0.0181{\mathbb{i}}}} \\ {{- 0.0056} + {0.3076{\mathbb{i}}}} & {0.3423 - {0.3074{\mathbb{i}}}} \end{matrix}$ ${V\; 2\left( {:{,{:{,43}}}} \right)} = \begin{matrix} {0.4422 + {0.0928{\mathbb{i}}}} & {{- 0.0860} - {0.2241{\mathbb{i}}}} \\ {{- 0.0137} - {0.2102{\mathbb{i}}}} & {{- 0.6015} + {0.2527{\mathbb{i}}}} \\ {0.2479 + {0.5606{\mathbb{i}}}} & {{- 0.4157} - {0.2206{\mathbb{i}}}} \\ {0.2479 + {0.5606{\mathbb{i}}}} & {0.1543 + {0.5211{\mathbb{i}}}} \end{matrix}$ ${V\; 2\left( {:{,{:{,44}}}} \right)} = \begin{matrix} {0.0337 + {0.6193{\mathbb{i}}}} & {{- 0.1081} + {0.4460{\mathbb{i}}}} \\ {{- 0.2280} - {0.1515{\mathbb{i}}}} & {{- 0.3567} - {0.5167{\mathbb{i}}}} \\ {0.2280 + {0.1515{\mathbb{i}}}} & {{- 0.2354} - {0.1914{\mathbb{i}}}} \\ {0.4621 + {0.5019{\mathbb{i}}}} & {{- 0.1985} - {0.5136{\mathbb{i}}}} \end{matrix}$

${V\; 2\left( {:{,{:{,45}}}} \right)} = \begin{matrix} 0.5000 & {- 0.5000} \\ 0.5000 & {0 + {0.5000{\mathbb{i}}}} \\ 0.5000 & 0.5000 \\ 0.5000 & {0 - {0.5000{\mathbb{i}}}} \end{matrix}$ ${V\; 2\left( {:{,{:{,46}}}} \right)} = \begin{matrix} {0.0918 + {0.3270{\mathbb{i}}}} & {{- 0.5060} + {0.1443{\mathbb{i}}}} \\ {0.4045 - {0.4815{\mathbb{i}}}} & {{- 0.4682} - {0.5571{\mathbb{i}}}} \\ {{- 0.1313} - {0.6387{\mathbb{i}}}} & {0.4345 + {0.0089{\mathbb{i}}}} \\ {0.0541 - {0.2473{\mathbb{i}}}} & {{- 0.0309} + {0.0618{\mathbb{i}}}} \end{matrix}$ ${V\; 2\left( {:{,{:{,47}}}} \right)} = \begin{matrix} {0.3841 + {0.3851{\mathbb{i}}}} & {0.0427 + {0.4037{\mathbb{i}}}} \\ {0.6580 - {0.2285{\mathbb{i}}}} & {{- 0.3221} + {0.3130{\mathbb{i}}}} \\ {{- 0.3448} - {0.0735{\mathbb{i}}}} & {{- 0.0724} + {0.1313{\mathbb{i}}}} \\ {0.3076 + {0.0056{\mathbb{i}}}} & {0.3259 - {0.7104{\mathbb{i}}}} \end{matrix}$ ${V\; 2\left( {:{,{:{,48}}}} \right)} = \begin{matrix} {0.4422 + {0.0928{\mathbb{i}}}} & {{- 0.1512} + {0.0220{\mathbb{i}}}} \\ {0.2102 - {0.0137{\mathbb{i}}}} & {{- 0.5297} - {0.2125{\mathbb{i}}}} \\ {{- 0.2479} - {0.5606{\mathbb{i}}}} & {0.3488 + {0.2574{\mathbb{i}}}} \\ {0.5606 - {0.2479{\mathbb{i}}}} & {0.5345 - {0.4211{\mathbb{i}}}} \end{matrix}$

${V\; 2\left( {:{,{:{,49}}}} \right)} = \begin{matrix} 0.5000 & {- 0.5000} \\ {0 + {0.5000{\mathbb{i}}}} & {- 0.5000} \\ 0.5000 & 0.5000 \\ {0 + {0.5000{\mathbb{i}}}} & 0.5000 \end{matrix}$ ${V\; 2\left( {:{,{:{,50}}}} \right)} = \begin{matrix} {0.3841 + {0.3851{\mathbb{i}}}} & {0.3899 + {0.5099{\mathbb{i}}}} \\ {0.0022 + {0.1690{\mathbb{i}}}} & {{- 0.2642} - {0.2730{\mathbb{i}}}} \\ {{- 0.3076} - {0.0056{\mathbb{i}}}} & {0.0634 + {0.5454{\mathbb{i}}}} \\ {{- 0.7536} - {0.1140{\mathbb{i}}}} & {0.3743 - {0.0464{\mathbb{i}}}} \end{matrix}$ ${V\; 2\left( {:{,{:{,51}}}} \right)} = \begin{matrix} {{- 0.1560} + {0.4926{\mathbb{i}}}} & {0.1996 + {0.3349{\mathbb{i}}}} \\ {0.0837 + {0.4175{\mathbb{i}}}} & {0.1647 + {0.4478{\mathbb{i}}}} \\ {{- 0.4597} + {0.3989{\mathbb{i}}}} & {0.0029 - {0.0825{\mathbb{i}}}} \\ {{- 0.4175} + {0.0837{\mathbb{i}}}} & {0.6245 - {0.4727{\mathbb{i}}}} \end{matrix}$ ${V\; 2\left( {:{,{:{,52}}}} \right)} = \begin{matrix} {0.3841 + {0.3851{\mathbb{i}}}} & {0.4945 - {0.2660{\mathbb{i}}}} \\ {{- 0.1140} + {0.7536{\mathbb{i}}}} & {{- 0.4531} - {0.0810{\mathbb{i}}}} \\ {{- 0.3076} - {0.0056{\mathbb{i}}}} & {0.2295 + {0.4493{\mathbb{i}}}} \\ {{- 0.1690} + {0.0022{\mathbb{i}}}} & {0.0234 - {0.4666{\mathbb{i}}}} \end{matrix}$

${V\; 2\left( {:{,{:{,53}}}} \right)} = \begin{matrix} 0.5000 & {- 0.5000} \\ {0 + {0.5000{\mathbb{i}}}} & 0.5000 \\ 0.5000 & 0.5000 \\ {0 + {0.5000{\mathbb{i}}}} & {- 0.5000} \end{matrix}$ ${V\; 2\left( {:{,{:{,54}}}} \right)} = \begin{matrix} {0.3396 + {0.1614{\mathbb{i}}}} & {{- 0.0891} - {0.1737{\mathbb{i}}}} \\ {{- 0.3400} - {0.3060{\mathbb{i}}}} & {{- 0.2160} + {0.0403{\mathbb{i}}}} \\ {0.3493 - {0.5641{\mathbb{i}}}} & {0.3530 + {0.5446{\mathbb{i}}}} \\ {{- 0.3060} + {0.3400{\mathbb{i}}}} & {0.1657 + {0.6818{\mathbb{i}}}} \end{matrix}$ ${V\; 2\left( {:{,{:{,55}}}} \right)} = \begin{matrix} {0.3841 + {0.3851{\mathbb{i}}}} & {0.3162 - {0.1330{\mathbb{i}}}} \\ {{- 0.1690} + {0.0022{\mathbb{i}}}} & {{- 0.2451} + {0.4651{\mathbb{i}}}} \\ {0.3076 + {0.0056{\mathbb{i}}}} & {0.1535 - {0.5047{\mathbb{i}}}} \\ {{- 0.1140} + {0.7536{\mathbb{i}}}} & {{- 0.4972} - {0.2836{\mathbb{i}}}} \end{matrix}$ ${V\; 2\left( {:{,{:{,56}}}} \right)} = \begin{matrix} {{- 0.1560} + {0.4926{\mathbb{i}}}} & {0.1157 + {0.3948{\mathbb{i}}}} \\ {{- 0.4175} + {0.0837{\mathbb{i}}}} & {0.2814 - {0.3175{\mathbb{i}}}} \\ {0.4597 - {0.3989{\mathbb{i}}}} & {0.4393 + {0.0781{\mathbb{i}}}} \\ {0.0837 + {0.4175{\mathbb{i}}}} & {0.3704 - {0.5609{\mathbb{i}}}} \end{matrix}$

${V\; 2\left( {:{,{:{,57}}}} \right)} = \begin{matrix} 0.5000 & 0.5000 \\ 0.5000 & {- 0.5000} \\ 0.5000 & 0.5000 \\ {- 0.5000} & 0.5000 \end{matrix}$ ${V\; 2\left( {:{,{:{,58}}}} \right)} = \begin{matrix} {0.3841 + {0.3851{\mathbb{i}}}} & {{- 0.6183} - {0.3893{\mathbb{i}}}} \\ {0.1140 - {0.7536{\mathbb{i}}}} & {0.0439 - {0.4301{\mathbb{i}}}} \\ {{- 0.3076} - {0.0056{\mathbb{i}}}} & {{- 0.3231} + {0.0445{\mathbb{i}}}} \\ {0.1690 - {0.0022{\mathbb{i}}}} & {{- 0.2461} - {0.3351{\mathbb{i}}}} \end{matrix}$ ${V\; 2\left( {:{,{:{,59}}}} \right)} = \begin{matrix} {0.3396 + {0.1614{\mathbb{i}}}} & {0.3566 - {0.2418{\mathbb{i}}}} \\ {0.3060 - {0.3400{\mathbb{i}}}} & {{- 0.1924} + {0.0246{\mathbb{i}}}} \\ {{- 0.3493} + {0.5641{\mathbb{i}}}} & {{- 0.5527} + {0.1626{\mathbb{i}}}} \\ {0.3400 + {0.3060{\mathbb{i}}}} & {{- 0.4030} - {0.5315{\mathbb{i}}}} \end{matrix}$ ${V\; 2\left( {:{,{:{,60}}}} \right)} = \begin{matrix} {0.3841 + {0.3851{\mathbb{i}}}} & {0.0072 + {0.6080{\mathbb{i}}}} \\ {{- 0.0022} - {0.1690{\mathbb{i}}}} & {{- 0.1531} - {0.3405{\mathbb{i}}}} \\ {{- 0.3076} - {0.0056{\mathbb{i}}}} & {0.4820 + {0.4643{\mathbb{i}}}} \\ {0.7536 + {0.1140{\mathbb{i}}}} & {{- 0.1738} - {0.1132{\mathbb{i}}}} \end{matrix}$

${V\; 2\left( {:{,{:{,61}}}} \right)} = \begin{matrix} 0.5000 & 0.5000 \\ {{- 0.3536} + {03536{\mathbb{i}}}} & {0.3536 - {0.3536{\mathbb{i}}}} \\ {0 - {0.5000{\mathbb{i}}}} & {0 - {0.5000{\mathbb{i}}}} \\ {0.3536 + {0.3536{\mathbb{i}}}} & {{- 0.3536} - {0.3536{\mathbb{i}}}} \end{matrix}$ ${V\; 2\left( {:{,{:{,62}}}} \right)} = \begin{matrix} {{- 0.1560} + {0.4926{\mathbb{i}}}} & {{- 0.4387} - {0.0343{\mathbb{i}}}} \\ {0.4175 - {0.0837{\mathbb{i}}}} & {0.5062 - {0.5972{\mathbb{i}}}} \\ {0.4597 - {0.3989{\mathbb{i}}}} & {{- 0.2611} + {0.2853{\mathbb{i}}}} \\ {{- 0.0837} - {0.4175{\mathbb{i}}}} & {{- 0.0591} + {0.2011{\mathbb{i}}}} \end{matrix}$ ${V\; 2\left( {:{,{:{,63}}}} \right)} = \begin{matrix} {0.3841 + {0.3851{\mathbb{i}}}} & {{- 0.0757} - {0.5813{\mathbb{i}}}} \\ {0.7536 + {0.1140{\mathbb{i}}}} & {0.1019 + {0.2454{\mathbb{i}}}} \\ {0.3076 + {0.0056{\mathbb{i}}}} & {0.5989 + {0.2606{\mathbb{i}}}} \\ {{- 0.0022} - {0.1690{\mathbb{i}}}} & {{- 0.3284} + {0.2264{\mathbb{i}}}} \end{matrix}$ ${V\; 2\left( {:{,{:{,64}}}} \right)} = \begin{matrix} {0.3396 + {0.1614{\mathbb{i}}}} & {0.2711 - {0.7070{\mathbb{i}}}} \\ {0.3400 + {0.3060{\mathbb{i}}}} & {{- 0.4123} - {0.1234{\mathbb{i}}}} \\ {0.3493 - {0.5641{\mathbb{i}}}} & {0.4714 - {0.0224{\mathbb{i}}}} \\ {0.3060 - {0.3400{\mathbb{i}}}} & {{- 0.0617} - {0.1224{\mathbb{i}}}} \end{matrix}$

-   -   Final Rank 3 Codebook:

${V\; 3\left( {:{,{:{,1}}}} \right)} = \begin{matrix} 0.5000 & 0.5000 & {- 0.5000} \\ 0.5000 & {- 0.5000} & {- 0.5000} \\ 0.5000 & 0.5000 & 0.5000 \\ 0.5000 & {- 0.5000} & 0.5000 \end{matrix}$ ${V\; 3\left( {:{,{:{,2}}}} \right)} = \begin{matrix} {0.4688 + {0.2170{\mathbb{i}}}} & {0.3008 + {0.0981{\mathbb{i}}}} & {0.1395 - {0.2201{\mathbb{i}}}} \\ {{- 0.2948} - {0.0700{\mathbb{i}}}} & {{- 0.1167} + {0.7005{\mathbb{i}}}} & {{- 0.4967} + {0.1489{\mathbb{i}}}} \\ {{- 0.4598} - {0.2965{\mathbb{i}}}} & {{- 0.1297} + {0.0716{\mathbb{i}}}} & {0.7351 + {0.0569{\mathbb{i}}}} \\ {{- 0.5846} - {0.0154{\mathbb{i}}}} & {0.3317 - {0.5135{\mathbb{i}}}} & {{- 0.3430} - {0.0437{\mathbb{i}}}} \end{matrix}$ ${V\; 3\left( {:{,{:{,3}}}} \right)} = \begin{matrix} {{- 0.4377} + {0.4498{\mathbb{i}}}} & {0.2079 + {0.3012{\mathbb{i}}}} & {{- 0.4172} - {0.5239{\mathbb{i}}}} \\ {{- 0.1173} + {0.2079{\mathbb{i}}}} & {0.4297 + {0.2345{\mathbb{i}}}} & {0.4687 + {0.3722{\mathbb{i}}}} \\ {{- 0.2098} + {0.2390{\mathbb{i}}}} & {{- 0.2835} - {0.7076{\mathbb{i}}}} & {{- 0.0338} - {0.0330{\mathbb{i}}}} \\ {0.6133 - {0.2682{\mathbb{i}}}} & {0.1669 + {0.1324{\mathbb{i}}}} & {{- 0.1311} - {0.4169{\mathbb{i}}}} \end{matrix}$ ${V\; 3\left( {:{,{:{,4}}}} \right)} = \begin{matrix} {0.4585 - {0.3521{\mathbb{i}}}} & {0.4797 - {0.5371{\mathbb{i}}}} & {0.1760 - {0.0285{\mathbb{i}}}} \\ {{- 0.5369} + {0.2492{\mathbb{i}}}} & {0.2256 + {0.0214{\mathbb{i}}}} & {0.0252 - {0.4154{\mathbb{i}}}} \\ {{- 0.1391} + {0.3154{\mathbb{i}}}} & {{- 0.1147} - {0.1411{\mathbb{i}}}} & {0.7276 + {0.5046{\mathbb{i}}}} \\ {0.2872 - {0.3378{\mathbb{i}}}} & {{- 0.2334} + {0.5852{\mathbb{i}}}} & {0.0121 + {0.1041{\mathbb{i}}}} \end{matrix}$

${V\; 3\left( {:{,{:{,5}}}} \right)} = \begin{matrix} 0.5000 & 0.5000 & {- 0.5000} \\ 0.5000 & {- 0.5000} & 0.5000 \\ 0.5000 & 0.5000 & 0.5000 \\ 0.5000 & {- 0.5000} & {- 0.5000} \end{matrix}$ ${V\; 3\left( {:{,{:{,6}}}} \right)} = \begin{matrix} {{- 0.7326} - {0.3303{\mathbb{i}}}} & {0.1180 - {0.0570{\mathbb{i}}}} & {0.3116 - {0.3531{\mathbb{i}}}} \\ {{- 0.1521} + {0.1245{\mathbb{i}}}} & {{- 0.3996} - {0.1865{\mathbb{i}}}} & {0.5845 - {0.0141{\mathbb{i}}}} \\ {0.0048 - {0.3893{\mathbb{i}}}} & {{- 0.8009} - {0.0266{\mathbb{i}}}} & {{- 0.3599} - {0.1123{\mathbb{i}}}} \\ {{- 0.3093} - {0.2615{\mathbb{i}}}} & {0.1996 - {0.3262{\mathbb{i}}}} & {{- 0.5417} + {0.0279{\mathbb{i}}}} \end{matrix}$ ${V\; 3\left( {:{,{:{,7}}}} \right)} = \begin{matrix} {{- 0.4259} - {0.6865{\mathbb{i}}}} & {0.1888 - {0.1148{\mathbb{i}}}} & {0.0505 - {0.0112{\mathbb{i}}}} \\ {0.0000 - {0.0163{\mathbb{i}}}} & {0.5849 + {0.0396{\mathbb{i}}}} & {0.1477 + {0.7346{\mathbb{i}}}} \\ {{- 0.0029} + {0.5625{\mathbb{i}}}} & {0.2943 + {0.2161{\mathbb{i}}}} & {0.2547 - {0.0108{\mathbb{i}}}} \\ {{- 0.1664} + {0.0538{\mathbb{i}}}} & {0.6828 - {0.0892{\mathbb{i}}}} & {{- 0.3558} - {0.4943{\mathbb{i}}}} \end{matrix}$ ${V\; 3\left( {:{,{:{,8}}}} \right)} = \begin{matrix} {{- 0.2522} + {0.2830{\mathbb{i}}}} & {0.0842 - {0.4310{\mathbb{i}}}} & {0.0964 + {0.2984{\mathbb{i}}}} \\ {{- 0.2511} + {0.3562{\mathbb{i}}}} & {{- 0.5001} - {0.4577{\mathbb{i}}}} & {{- 0.4236} - {0.1898{\mathbb{i}}}} \\ {{- 0.6354} - {0.2810{\mathbb{i}}}} & {{- 0.3335} + {0.3466{\mathbb{i}}}} & {{- 0.1484} - {0.2414{\mathbb{i}}}} \\ {{- 0.2798} - {0.3246{\mathbb{i}}}} & {{- 0.1512} - {0.1513{\mathbb{i}}}} & {0.0306 + {0.7778{\mathbb{i}}}} \end{matrix}$

${V\; 3\left( {:{,{:{,9}}}} \right)} = \begin{matrix} 0.5000 & {- 0.5000} & {- 0.5000} \\ 0.5000 & {- 0.5000} & 0.5000 \\ 0.5000 & 0.5000 & 0.5000 \\ 0.5000 & 0.5000 & {- 0.5000} \end{matrix}$ ${V\; 3\left( {:{,{:{,10}}}} \right)} = \begin{matrix} {0.4517 - {0.1172{\mathbb{i}}}} & {{- 0.0439} + {0.5591{\mathbb{i}}}} & {0.3594 - {0.4725{\mathbb{i}}}} \\ {{- 0.3568} + {0.3992{\mathbb{i}}}} & {{- 0.4696} - {0.0932{\mathbb{i}}}} & {{- 0.2384} - {0.6026{\mathbb{i}}}} \\ {0.0918 + {0.4242{\mathbb{i}}}} & {0.4085 - {0.0481{\mathbb{i}}}} & {0.3325 - {0.3269{\mathbb{i}}}} \\ {{- 0.3837} - {0.3999{\mathbb{i}}}} & {{- 0.3005} + {0.4437{\mathbb{i}}}} & {{- 0.0155} - {0.1000{\mathbb{i}}}} \end{matrix}$ ${V\; 3\left( {:{,{:{,11}}}} \right)} = \begin{matrix} {0.3983 + {0.5597{\mathbb{i}}}} & {{- 0.1922} - {0.0358{\mathbb{i}}}} & {{- 0.2096} + {0.2475{\mathbb{i}}}} \\ {{- 0.5583} + {0.0724{\mathbb{i}}}} & {0.2420 - {0.0149{\mathbb{i}}}} & {0.3931 - {0.0653{\mathbb{i}}}} \\ {0.2543 + {0.2747{\mathbb{i}}}} & {0.7756 - {0.0378{\mathbb{i}}}} & {0.3367 + {0.2619{\mathbb{i}}}} \\ {{- 0.0069} + {0.2664{\mathbb{i}}}} & {{- 0.5242} + {0.1589{\mathbb{i}}}} & {0.7417 + {0.0629{\mathbb{i}}}} \end{matrix}$ ${V\; 3\left( {:{,{:{,12}}}} \right)} = \begin{matrix} {0.3609 + {0.0743{\mathbb{i}}}} & {{- 0.6309} - {0.1183{\mathbb{i}}}} & {0.2159 - {0.4488{\mathbb{i}}}} \\ {{- 0.0447} - {0.2624{\mathbb{i}}}} & {{- 0.3156} + {0.2879{\mathbb{i}}}} & {{- 0.0988} + {0.6010{\mathbb{i}}}} \\ {0.2646 - {0.2157{\mathbb{i}}}} & {{- 0.3493} - {0.1821{\mathbb{i}}}} & {0.3113 + {0.5056{\mathbb{i}}}} \\ {{- 0.7438} + {0.3516{\mathbb{i}}}} & {{- 0.4975} - {0.0539{\mathbb{i}}}} & {{- 0.1643} + {0.0375{\mathbb{i}}}} \end{matrix}$

${V\; 3\left( {:{,{:{,13}}}} \right)} = \begin{matrix} 0.5000 & {- 0.5000} & {- 0.5000} \\ {- 0.5000} & {- 0.5000} & 0.5000 \\ 0.5000 & 0.5000 & 0.5000 \\ {- 0.5000} & 0.5000 & {- 0.5000} \end{matrix}$ ${V\; 3\left( {:{,{:{,14}}}} \right)} = \begin{matrix} {{- 0.4197} + {0.4817{\mathbb{i}}}} & {0.0820 + {0.2932{\mathbb{i}}}} & {0.5394 + {0.0634{\mathbb{i}}}} \\ {{- 0.2858} - {0.0562{\mathbb{i}}}} & {0.0717 - {0.5527{\mathbb{i}}}} & {0.3834 + {0.2860{\mathbb{i}}}} \\ {{- 0.0712} + {0.2154{\mathbb{i}}}} & {{- 0.2709} + {0.6053{\mathbb{i}}}} & {{- 0.2652} + {0.2505{\mathbb{i}}}} \\ {0.1024 + {0.6671{\mathbb{i}}}} & {{- 0.0386} - {0.3943{\mathbb{i}}}} & {{- 0.3941} + {0.4333{\mathbb{i}}}} \end{matrix}$ ${V\; 3\left( {:{,{:{,15}}}} \right)} = \begin{matrix} {0.3849 + {0.1986{\mathbb{i}}}} & {{- 0.5420} - {0.2820{\mathbb{i}}}} & {0.3395 + {0.1672{\mathbb{i}}}} \\ {{- 0.1824} - {0.0094{\mathbb{i}}}} & {0.1989 + {0.1166{\mathbb{i}}}} & {0.6694 + {0.6089{\mathbb{i}}}} \\ {0.2254 + {0.6795{\mathbb{i}}}} & {{- 0.1979} + {0.5597{\mathbb{i}}}} & {{- 0.0690} + {0.0775{\mathbb{i}}}} \\ {{- 0.5154} + {0.0292{\mathbb{i}}}} & {0.0915 + {0.4612{\mathbb{i}}}} & {0.1367 - {0.0921{\mathbb{i}}}} \end{matrix}$ ${V\; 3\left( {:{,{:{,16}}}} \right)} = \begin{matrix} {0.6638 - {0.4917{\mathbb{i}}}} & {{- 0.0134} - {0.1109{\mathbb{i}}}} & {0.3780 + {0.2166{\mathbb{i}}}} \\ {{- 0.1566} - {0.1262{\mathbb{i}}}} & {0.0722 - {0.9308{\mathbb{i}}}} & {{- 0.1132} + {0.1055{\mathbb{i}}}} \\ {{- 0.0029} - {0.3366{\mathbb{i}}}} & {{- 0.1758} + {0.1916{\mathbb{i}}}} & {{- 0.1479} + {0.6100{\mathbb{i}}}} \\ {{- 0.3978} + {0.0753{\mathbb{i}}}} & {0.2200 - {0.0040{\mathbb{i}}}} & {{- 0.0224} + {0.6259{\mathbb{i}}}} \end{matrix}$

${V\; 3\left( {:{,{:{,17}}}} \right)} = \begin{matrix} 0.5000 & 0.5000 & {- 0.5000} \\ {0 + {0.5000{\mathbb{i}}}} & {0 - {0.5000{\mathbb{i}}}} & {0 - {0.5000{\mathbb{i}}}} \\ 0.5000 & 0.5000 & 0.5000 \\ {0 + {0.5000{\mathbb{i}}}} & {0 - {0.5000{\mathbb{i}}}} & {0 + {0.5000{\mathbb{i}}}} \end{matrix}$ ${V\; 3\left( {:{,{:{,18}}}} \right)} = \begin{matrix} {0.2420 - {0.2055{\mathbb{i}}}} & {{- 0.1388} + {0.6904{\mathbb{i}}}} & {0.3273 + {0.0199{\mathbb{i}}}} \\ {{- 0.5655} - {0.6754{\mathbb{i}}}} & {{- 0.2104} + {0.2029{\mathbb{i}}}} & {{- 0.1634} + {0.1312{\mathbb{i}}}} \\ {0.0981 + {0.0467{\mathbb{i}}}} & {0.1060 + {0.1310{\mathbb{i}}}} & {{- 0.7933} + {0.4542{\mathbb{i}}}} \\ {0.3298 - {0.0517{\mathbb{i}}}} & {0.5748 + {0.2447{\mathbb{i}}}} & {{- 0.0516} + {0.1018{\mathbb{i}}}} \end{matrix}$ ${V\; 3\left( {:{,{:{,19}}}} \right)} = \begin{matrix} {{- 0.2841} - {0.7969{\mathbb{i}}}} & {0.3713 - {0.0745{\mathbb{i}}}} & {0.0639 + {0.1460{\mathbb{i}}}} \\ {0.0227 + {0.2058{\mathbb{i}}}} & {0.5630 - {0.0230{\mathbb{i}}}} & {{- 0.0999} - {0.7520{\mathbb{i}}}} \\ {0.2505 + {0.3816{\mathbb{i}}}} & {0.1527 - {0.2390{\mathbb{i}}}} & {0.3715 + {0.3849{\mathbb{i}}}} \\ {{- 0.0898} - {0.1576{\mathbb{i}}}} & {{- 0.6437} - {0.2105{\mathbb{i}}}} & {0.0144 - {0.3358{\mathbb{i}}}} \end{matrix}$ ${V\; 3\left( {:{,{:{,20}}}} \right)} = \begin{matrix} {0.3344 - {0.1640{\mathbb{i}}}} & {{- 0.2485} + {0.4791{\mathbb{i}}}} & {0.4000 + {0.1590{\mathbb{i}}}} \\ {{- 0.1475} + {0.2641{\mathbb{i}}}} & {0.5893 + {0.1330{\mathbb{i}}}} & {{- 0.1383} - {0.2427{\mathbb{i}}}} \\ {{- 0.1383} + {0.3182{\mathbb{i}}}} & {{- 0.4811} + {0.3240{\mathbb{i}}}} & {{- 0.1561} - {0.4493{\mathbb{i}}}} \\ {{- 0.3105} - {0.7437{\mathbb{i}}}} & {{- 0.0811} - {0.0251{\mathbb{i}}}} & {{- 0.4817} + {0.1909{\mathbb{i}}}} \end{matrix}$

${V\; 3\left( {:{,{:{,21}}}} \right)} = \begin{matrix} 0.5000 & 0.5000 & {- 0.5000} \\ {0 + {0.5000{\mathbb{i}}}} & {0 - {0.5000{\mathbb{i}}}} & {0 + {0.5000{\mathbb{i}}}} \\ 0.5000 & 0.5000 & 0.5000 \\ {0 + {0.5000{\mathbb{i}}}} & {0 - {0.5000{\mathbb{i}}}} & {0 - {0.5000{\mathbb{i}}}} \end{matrix}$ ${V\; 3\left( {:{,{:{,22}}}} \right)} = \begin{matrix} {0.4332 + {0.1612{\mathbb{i}}}} & {{- 0.4077} - {0.0486{\mathbb{i}}}} & {0.4750 + {0.0865{\mathbb{i}}}} \\ {{- 0.2499} + {0.4647{\mathbb{i}}}} & {{- 0.3727} - {0.1094{\mathbb{i}}}} & {0.2747 + {0.1727{\mathbb{i}}}} \\ {0.1294 + {0.5446{\mathbb{i}}}} & {0.6827 - {0.1403{\mathbb{i}}}} & {0.1161 - {0.3354{\mathbb{i}}}} \\ {0.3535 - {0.2640{\mathbb{i}}}} & {0.4328 + {0.0863{\mathbb{i}}}} & {0.3135 + {0.6613{\mathbb{i}}}} \end{matrix}$ ${V\; 3\left( {:{,{:{,23}}}} \right)} = \begin{matrix} {0.5948 - {0.2086{\mathbb{i}}}} & {{- 0.6155} + {0.0029{\mathbb{i}}}} & {0.0246 + {0.1384{\mathbb{i}}}} \\ {{- 0.6101} + {0.1811{\mathbb{i}}}} & {{- 0.2226} - {0.2992{\mathbb{i}}}} & {0.0747 + {0.2731{\mathbb{i}}}} \\ {{- 0.1244} + {0.3625{\mathbb{i}}}} & {0.0659 + {0.6219{\mathbb{i}}}} & {0.2542 + {0.1473{\mathbb{i}}}} \\ {{- 0.1261`} + {0.1870{\mathbb{i}}}} & {{- 0.1866} + {0.2370{\mathbb{i}}}} & {{- 0.7184} - {0.5456{\mathbb{i}}}} \end{matrix}$ ${V\; 3\left( {:{,{:{,24}}}} \right)} = \begin{matrix} {{- 0.7276} + {0.3231{\mathbb{i}}}} & {{- 0.0746} + {0.0002{\mathbb{i}}}} & {0.1968 + {0.1617{\mathbb{i}}}} \\ {0.0623 - {0.3109{\mathbb{i}}}} & {0.7114 + {0.3826{\mathbb{i}}}} & {0.3902 + {0.0102{\mathbb{i}}}} \\ {{- 0.0316} - {0.3607{\mathbb{i}}}} & {{- 0.4356} + {0.1774{\mathbb{i}}}} & {0.0838 - {0.7186{\mathbb{i}}}} \\ {0.1496 - {0.3350{\mathbb{i}}}} & {{- 0.1859} - {0.2936{\mathbb{i}}}} & {{- 0.1918} + {0.4718{\mathbb{i}}}} \end{matrix}$

${V\; 3\left( {:{,{:{,25}}}} \right)} = \begin{matrix} 0.5000 & {- 0.5000} & {- 0.5000} \\ {0 + {0.5000{\mathbb{i}}}} & {0 - {0.5000{\mathbb{i}}}} & {0 + {0.5000{\mathbb{i}}}} \\ 0.5000 & 0.5000 & 0.5000 \\ {0 + {0.5000{\mathbb{i}}}} & {0 + {0.5000{\mathbb{i}}}} & {0 - {0.5000{\mathbb{i}}}} \end{matrix}$ ${V\; 3\left( {:{,{:{,26}}}} \right)} = \begin{matrix} {0.2560 + {0.1831{\mathbb{i}}}} & {0.2046 + {0.4515{\mathbb{i}}}} & {0.2200 + {0.5535{\mathbb{i}}}} \\ {0.0741 + {0.1142{\mathbb{i}}}} & {{- 0.2010} + {0.6413{\mathbb{i}}}} & {0.4491 - {0.4832{\mathbb{i}}}} \\ {0.3746 + {0.4660{\mathbb{i}}}} & {0.2087 + {0.1370{\mathbb{i}}}} & {{- 0.2423} + {0.1904{\mathbb{i}}}} \\ {{- 0.0601} - {0.7188{\mathbb{i}}}} & {{- 0.1085} + {0.4782{\mathbb{i}}}} & {{- 0.2656} + {0.2110{\mathbb{i}}}} \end{matrix}$ ${V\; 3\left( {:{,{:{,27}}}} \right)} = \begin{matrix} {0.4170 + {0.2673{\mathbb{i}}}} & {0.0729 - {0.3667{\mathbb{i}}}} & {{- 0.1258} + {0.1841{\mathbb{i}}}} \\ {{- 0.0605} + {0.0504{\mathbb{i}}}} & {{- 0.2440} + {0.1836{\mathbb{i}}}} & {0.8489 + {0.2116{\mathbb{i}}}} \\ {0.3145 + {0.3209{\mathbb{i}}}} & {{- 0.6599} - {0.3142{\mathbb{i}}}} & {{- 0.0292} - {0.3579{\mathbb{i}}}} \\ {{- 0.4946} + {0.5495{\mathbb{i}}}} & {{- 0.3770} + {0.3011{\mathbb{i}}}} & {{- 0.2007} + {0.1251{\mathbb{i}}}} \end{matrix}$ ${V\; 3\left( {:{,{:{,28}}}} \right)} = \begin{matrix} {{- 0.1568} + {0.5108{\mathbb{i}}}} & {{- 0.6900} + {0.1066{\mathbb{i}}}} & {0.4419 + {0.0901{\mathbb{i}}}} \\ {{- 0.2476} + {0.2774{\mathbb{i}}}} & {0.3823 - {0.1521{\mathbb{i}}}} & {0.3815 - {0.4492{\mathbb{i}}}} \\ {0.1736 - {0.2279{\mathbb{i}}}} & {{- 0.1974} - {0.4946{\mathbb{i}}}} & {0.3241 - {0.4623{\mathbb{i}}}} \\ {{- 0.6443} - {0.2811{\mathbb{i}}}} & {0.0584 + {0.2373{\mathbb{i}}}} & {0.0318 - {0.3599{\mathbb{i}}}} \end{matrix}$

${V\; 3\left( {:{,{:{,29}}}} \right)} = \begin{matrix} 0.5000 & {- 0.5000} & {- 0.5000} \\ {0 - {0.5000{\mathbb{i}}}} & {0. - {5000{\mathbb{i}}}} & {0 + {0.5000{\mathbb{i}}}} \\ 0.5000 & 0.5000 & 0.5000 \\ {0 - {0.5000{\mathbb{i}}}} & {0 + {0.5000{\mathbb{i}}}} & {0 - {0.5000{\mathbb{i}}}} \end{matrix}$ ${V\; 3\left( {:{,{:{,30}}}} \right)} = \begin{matrix} {{- 0.4915} + {0.5944{\mathbb{i}}}} & {0.0644 - {0.0145{\mathbb{i}}}} & {{- 0.1689} - {0.5903{\mathbb{i}}}} \\ {{- 0.1046} - {0.3964{\mathbb{i}}}} & {{- 0.5642} + {0.0031{\mathbb{i}}}} & {{- 0.3416} - {0.2274{\mathbb{i}}}} \\ {0.2288 + {0.3710{\mathbb{i}}}} & {{- 0.2561} - {0.0135{\mathbb{i}}}} & {0.6471 - {0.0998{\mathbb{i}}}} \\ {{- 0.2089} + {0.0583{\mathbb{i}}}} & {{- 0.4070} - {0.6677{\mathbb{i}}}} & {{- 0.1176} + {0.1097{\mathbb{i}}}} \end{matrix}$ ${V\; 3\left( {:{,{:{,31}}}} \right)} = \begin{matrix} {0.0849 - {0.1823{\mathbb{i}}}} & {0.5488 + {0.2367{\mathbb{i}}}} & {{- 0.3534} - {0.6018{\mathbb{i}}}} \\ {{- 0.1210} - {0.0233{\mathbb{i}}}} & {{- 0.2347} - {0.3498{\mathbb{i}}}} & {0.3248 - {0.5261{\mathbb{i}}}} \\ {0.4127 - {0.7969{\mathbb{i}}}} & {{- 0.2180} - {0.2794{\mathbb{i}}}} & {{- 0.0107} - {0.0700{\mathbb{i}}}} \\ {{- 0.0136} + {0.3727{\mathbb{i}}}} & {{- 0.3369} - {0.4757{\mathbb{i}}}} & {{- 0.2087} - {0.2866{\mathbb{i}}}} \end{matrix}$ ${V\; 3\left( {:{,{:{,32}}}} \right)} = \begin{matrix} {0.4828 - {0.2545{\mathbb{i}}}} & {0.2119 + {0.3871{\mathbb{i}}}} & {{- 0.3824} - {0.2556{\mathbb{i}}}} \\ {{- 0.1939} + {0.1069{\mathbb{i}}}} & {{- 0.3495} + {0.6691{\mathbb{i}}}} & {0.1365 + {0.5175{\mathbb{i}}}} \\ {0.3116 + {0.2702{\mathbb{i}}}} & {{- 0.1220} + {0.2056{\mathbb{i}}}} & {{- 0.5361} - {0.0030{\mathbb{i}}}} \\ {{- 0.4258} + {0.5492{\mathbb{i}}}} & {0.4208 + {0.0338{\mathbb{i}}}} & {{- 0.4460} + {0.1251{\mathbb{i}}}} \end{matrix}$

${V\; 3\left( {:{,{:{,33}}}} \right)} = \begin{matrix} 0.5000 & 0.5000 & {- 0.5000} \\ 0.5000 & {- 0.5000} & {0 - {0.5000{\mathbb{i}}}} \\ 0.5000 & 0.5000 & 0.5000 \\ 0.5000 & {- 0.5000} & {0 + {0.5000{\mathbb{i}}}} \end{matrix}$ ${V\; 3\left( {:{,{:{,34}}}} \right)} = \begin{matrix} {{- 0.3283} - {0.4339{\mathbb{i}}}} & {{- 0.2690} + {0.1335{\mathbb{i}}}} & {{- 0.4536} - {0.1528{\mathbb{i}}}} \\ {{- 0.3886} - {0.2987{\mathbb{i}}}} & {0.0982 - {0.6284{\mathbb{i}}}} & {0.5255 + {0.0653{\mathbb{i}}}} \\ {0.2457 - {0.23371{\mathbb{i}}}} & {{- 0.3618} + {0.5436{\mathbb{i}}}} & {0.6122 + {0.0946{\mathbb{i}}}} \\ {{- 0.4861} - {0.3354{\mathbb{i}}}} & {{- 0.2060} + {0.1911{\mathbb{i}}}} & {0.0216 + {0.3261{\mathbb{i}}}} \end{matrix}$ ${V\; 3\left( {:{,{:{,35}}}} \right)} = \begin{matrix} {{- 0.0348} + {0.4484{\mathbb{i}}}} & {{- 0.4669} - {0.6204{\mathbb{i}}}} & {0.2685 + {0.0858{\mathbb{i}}}} \\ {{- 0.4400} + {0.2694{\mathbb{i}}}} & {0.0511 + {0.4386{\mathbb{i}}}} & {{- 0.1700} + {0.3385{\mathbb{i}}}} \\ {0.3429 - {0.0049{\mathbb{i}}}} & {0.4475 - {0.0066{\mathbb{i}}}} & {0.1480 - {0.4849{\mathbb{i}}}} \\ {0.1991 + {0.6118{\mathbb{i}}}} & {0.0419 + {0.0073{\mathbb{i}}}} & {{- 0.6988} - {0.1782{\mathbb{i}}}} \end{matrix}$ ${V\; 3\left( {:{,{:{,36}}}} \right)} = \begin{matrix} {{- 0.2360} - {0.2973{\mathbb{i}}}} & {0.3316 + {0.2303{\mathbb{i}}}} & {0.0526 + {0.6279{\mathbb{i}}}} \\ {0.0315 - {0.2366{\mathbb{i}}}} & {{- 0.3698} - {0.2862{\mathbb{i}}}} & {0.3197 - {0.3701{\mathbb{i}}}} \\ {0.7285 + {0.0791{\mathbb{i}}}} & {0.5225 + {0.0720{\mathbb{i}}}} & {{- 0.0255} - {0.2448{\mathbb{i}}}} \\ {0.4097 - {0.3067{\mathbb{i}}}} & {{- 0.4142} + {0.4106{\mathbb{i}}}} & {0.5048 + {0.2198{\mathbb{i}}}} \end{matrix}$

${V\; 3\left( {:{,{:{,37}}}} \right)} = \begin{matrix} 0.5000 & {- 0.5000} & {- 0.5000} \\ 0.5000 & {0 - {0.5000{\mathbb{i}}}} & {0 + {0.5000{\mathbb{i}}}} \\ 0.5000 & 0.5000 & 0.5000 \\ 0.5000 & {0 + {0.5000{\mathbb{i}}}} & {0 - {0.5000{\mathbb{i}}}} \end{matrix}$ ${V\; 3\left( {:{,{:{,38}}}} \right)} = \begin{matrix} {{- 0.5245} - {0.3774{\mathbb{i}}}} & {{- 0.5623} - {0.0655{\mathbb{i}}}} & {0.2278 + {0.0771{\mathbb{i}}}} \\ {{- 0.0522} + {0.4805{\mathbb{i}}}} & {{- 0.1132} - {0.5413{\mathbb{i}}}} & {0.6451 - {0.0046{\mathbb{i}}}} \\ {0.1023 + {0.0593{\mathbb{i}}}} & {{- 0.5232} - {0.2730{\mathbb{i}}}} & {{- 0.4236} + {0.2873{\mathbb{i}}}} \\ {{- 0.5756} - {0.0591{\mathbb{i}}}} & {0.0088 - {0.1592{\mathbb{i}}}} & {{- 0.3254} - {0.3977{\mathbb{i}}}} \end{matrix}$ ${V\; 3\left( {:{,{:{,39}}}} \right)} = \begin{matrix} {0.1953 - {0.3057{\mathbb{i}}}} & {{- 0.5062} + {0.0964{\mathbb{i}}}} & {0.4500 + {0.1253{\mathbb{i}}}} \\ {{- 0.2685} - {0.1895{\mathbb{i}}}} & {0.6204 + {0.3044{\mathbb{i}}}} & {0.1623 + {0.5596{\mathbb{i}}}} \\ {{- 0.7779} - {0.3721{\mathbb{i}}}} & {{- 0.3442} - {0.2479{\iota i}}} & {{- 0.0309} + {0.0263{\mathbb{i}}}} \\ {{- 0.0216} - {0.1283{\mathbb{i}}}} & {0.2771 + {0.0109{\mathbb{i}}}} & {{- 0.1897} - {0.6361{\mathbb{i}}}} \end{matrix}$ ${V\; 3\left( {:{,{:{,40}}}} \right)} = \begin{matrix} {{- 0.6035} - {0.3342{\mathbb{i}}}} & {{- 0.3380} - {0.3706{\mathbb{i}}}} & {{- 0.0686} + {0.3905{\mathbb{i}}}} \\ {{- 0.1428} + {0.4838{\mathbb{i}}}} & {{- 0.1430} - {0.1006{\mathbb{i}}}} & {{- 0.3957} - {0.4036{\mathbb{i}}}} \\ {{- 0.1927} + {0.1626{\mathbb{i}}}} & {{- 0.0874} + {0.0678{\mathbb{i}}}} & {{- 0.6478} + {0.2818{\mathbb{i}}}} \\ {{- 0.3638} - {0.2714{\mathbb{i}}}} & {0.3209 + {0.7763{\mathbb{i}}}} & {{- 0.1557} - {0.0019{\mathbb{i}}}} \end{matrix}$

${V\; 3\left( {:{,{:{,41}}}} \right)} = \begin{matrix} 0.5000 & 0.5000 & {- 0.5000} \\ {0 + {0.5000{\mathbb{i}}}} & {0 - {0.5000{\mathbb{i}}}} & {- 0.5000} \\ 0.5000 & 0.5000 & 0.5000 \\ {0 + {0.5000{\mathbb{i}}}} & {0 - {0.5000{\mathbb{i}}}} & 0.5000 \end{matrix}$ ${V\; 3\left( {:{,{:{,42}}}} \right)} = \begin{matrix} {0.1371 + {0.7838{\mathbb{i}}}} & {0.2479 + {0.0123{\mathbb{i}}}} & {{- 0.0008} - {0.0975{\mathbb{i}}}} \\ {{- 0.0956} + {0.3657{\mathbb{i}}}} & {0.2440 + {0.34471{\mathbb{i}}}} & {{- 0.4391} - {0.0271{\mathbb{i}}}} \\ {{- 0.1099} - {0.0181{\mathbb{i}}}} & {{- 0.2141} + {0.1937{\mathbb{i}}}} & {{- 0.2988} + {0.8311{\mathbb{i}}}} \\ {0.3423 - {0.3074{\mathbb{i}}}} & {0.2247 + {0.7913{\mathbb{i}}}} & {{- 0.0349} - {0.1254{\mathbb{i}}}} \end{matrix}$ ${V\; 3\left( {:{,{:{,43}}}} \right)} = \begin{matrix} {{- 0.0860} - {0.2241{\mathbb{i}}}} & {0.5338 - {0.0441{\mathbb{i}}}} & {{- 0.1033} + {0.6639{\mathbb{i}}}} \\ {{- 0.6015} + {0.2527{\mathbb{i}}}} & {{- 0.4372} + {0.3957{\mathbb{i}}}} & {{- 0.1471} + {0.4008{\mathbb{i}}}} \\ {{- 0.4157} - {0.2206{\mathbb{i}}}} & {0.0012 + {0.2819{\mathbb{i}}}} & {0.4969 - {0.2768{\mathbb{i}}}} \\ {0.1543 + {0.5211{\mathbb{i}}}} & {{- 0.3515} - {0.4030{\mathbb{i}}}} & {{- 0.0551} + {0.1997{\mathbb{i}}}} \end{matrix}$ ${V\; 3\left( {:{,{:{,44}}}} \right)} = \begin{matrix} {{- 0.1081} + {0.4460{\mathbb{i}}}} & {0.0435 - {0.3239{\mathbb{i}}}} & {{- 0.1491} + {0.5251{\mathbb{i}}}} \\ {{- 0.3567} - {0.5167{\mathbb{i}}}} & {{- 0.1969} - {0.3585{\mathbb{i}}}} & {0.3858 + {0.4634{\mathbb{i}}}} \\ {{- 0.2354} - {0.1914{\mathbb{i}}}} & {0.2532 + {0.7745{\mathbb{i}}}} & {0.0863 + {0.4020{\mathbb{i}}}} \\ {{- 0.1985} - {0.5136{\mathbb{i}}}} & {0.0895 - {0.2322{\mathbb{i}}}} & {{- 0.0884} - {0.4019{\mathbb{i}}}} \end{matrix}$

${V\; 3\left( {:{,{:{,45}}}} \right)} = \begin{matrix} 0.5000 & {- 0.5000} & {- 0.5000} \\ {0 + {0.5000{\mathbb{i}}}} & {- 0.5000} & 0.5000 \\ 0.5000 & 0.5000 & 0.5000 \\ {0 + {0.5000{\mathbb{i}}}} & 0.5000 & {- 0.5000} \end{matrix}$ ${V\; 3\left( {:{,{:{,46}}}} \right)} = \begin{matrix} {{- 0.5060} + {0.1443{\mathbb{i}}}} & {0.5534 - {0.3836{\mathbb{i}}}} & {{- 0.1155} - {0.3757{\mathbb{i}}}} \\ {{- 0.4682} - {0.5571{\mathbb{i}}}} & {{- 0.1369} - {0.0572{\mathbb{i}}}} & {{- 0.2290} + {0.0234{\mathbb{i}}}} \\ {0.4345 + {0.0089{\mathbb{i}}}} & {0.5727 - {0.1978{\mathbb{i}}}} & {{- 0.1012} - {0.0932{\mathbb{i}}}} \\ {{- 0.0309} + {0.0618{\iota i}}} & {{- 0.2903} - {0.2707{\mathbb{i}}}} & {0.7177 - {0.5085{\mathbb{i}}}} \end{matrix}$ ${V\; 3\left( {:{,{:{,47}}}} \right)} = \begin{matrix} {0.0427 + {0.4037{\mathbb{i}}}} & {0.1038 + {0.3577{\mathbb{i}}}} & {0.5172 + {0.3648{\mathbb{i}}}} \\ {{- 0.3221} + {0.3130{\mathbb{i}}}} & {{- 0.3384} - {0.3674{\mathbb{i}}}} & {{- 0.2477} + {0.0474{\mathbb{i}}}} \\ {{- 0.0724} + {0.1313{\mathbb{i}}}} & {{- 0.0231} - {0.7092{\mathbb{i}}}} & {0.5412 + {0.2385{\mathbb{i}}}} \\ {0.3259 - {0.7104{\mathbb{i}}}} & {{- 0.3205} - {0.0747{\mathbb{i}}}} & {0.1184 + {0.4148{\mathbb{i}}}} \end{matrix}$ ${V\; 3\left( {:{,{:{,48}}}} \right)} = \begin{matrix} {{- 0.1512} + {0.0220{\mathbb{i}}}} & {{- 0.5578} + {0.3060{\mathbb{i}}}} & {{- 0.4803} + {0.4790{\mathbb{i}}}} \\ {{- 0.5297} - {0.2125{\mathbb{i}}}} & {{- 0.4245} + {0.2405{\mathbb{i}}}} & {0.3258 - {0.5345{\mathbb{i}}}} \\ {0.3488 + {0.2574{\mathbb{i}}}} & {{- 0.5660} - {0.2718{\mathbb{i}}}} & {{- 0.0021} - {0.2051{\mathbb{i}}}} \\ {0.5345 - {0.4211{\mathbb{i}}}} & {0.1552 + {0.1767{\mathbb{i}}}} & {{- 0.1657} - {0.2802{\mathbb{i}}}} \end{matrix}$

${V\; 3\left( {:{,{:{,49}}}} \right)} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 \\ 0.5000 & {0 + {0.5000{\mathbb{i}}}} & {- 0.5000} \\ 0.5000 & {- 0.5000} & 0.5000 \\ {- 0.5000} & {0 + {0.5000{\mathbb{i}}}} & 0.5000 \end{matrix}$ ${V\; 3\left( {:{,{:{,50}}}} \right)} = \begin{matrix} {0.3899 + {0.5099{\mathbb{i}}}} & {0.1239 + {0.3480{\mathbb{i}}}} & {{- 0.3212} + {0.2294{\mathbb{i}}}} \\ {{- 0.2642} - {0.2730{\mathbb{i}}}} & {{- 0.0709} + {0.7215{\mathbb{i}}}} & {0.4401 + {0.3283{\mathbb{i}}}} \\ {0.0634 + {0.5454{\mathbb{i}}}} & {{- 0.0400} - {0.3119{\mathbb{i}}}} & {0.5426 + {0.4590{\mathbb{i}}}} \\ {0.3743 - {0.0464{\mathbb{i}}}} & {0.3735 + {0.3156{\mathbb{i}}}} & {{- 0.1921} - {0.0291{\mathbb{i}}}} \end{matrix}$ ${V\; 3\left( {:{,{:{,51}}}} \right)} = \begin{matrix} {0.1996 + {0.3349{\mathbb{i}}}} & {0.3567 - {0.6608{\mathbb{i}}}} & {0.1085 + {0.0728{\mathbb{i}}}} \\ {0.1647 + {0.4478{\mathbb{i}}}} & {{- 0.0991} + {0.3844{\mathbb{i}}}} & {{- 0.5490} - {0.3633{\mathbb{i}}}} \\ {0.0029 - {0.0825{\mathbb{i}}}} & {{- 0.4340} + {0.1941{\mathbb{i}}}} & {{- 0.0351} + {0.6288{\mathbb{i}}}} \\ {0.6245 - {0.4727{\mathbb{i}}}} & {0.1497 + {0.1732{\mathbb{i}}}} & {0.1381 - {0.3657{\mathbb{i}}}} \end{matrix}$ ${V\; 3\left( {:{,{:{,52}}}} \right)} = \begin{matrix} {0.4945 - {0.2660{\mathbb{i}}}} & {{- 0.4949} + {0.3789{\mathbb{i}}}} & {{- 0.0039} - {0.0184{\mathbb{i}}}} \\ {{- 0.4531} - {0.0810{\mathbb{i}}}} & {0.2332 + {0.1714{\mathbb{i}}}} & {{- 0.0493} - {0.3478{\mathbb{i}}}} \\ {0.2295 + {0.4493{\mathbb{i}}}} & {0.2178 + {0.6234{\mathbb{i}}}} & {{- 0.4191} + {0.1977{\mathbb{i}}}} \\ {0.0234 - {0.4666{\mathbb{i}}}} & {{- 0.0755} - {0.2931{\mathbb{i}}}} & {{- 0.8112} + {0.0581{\mathbb{i}}}} \end{matrix}$

${V\; 3\left( {:{,{:{,53}}}} \right)} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 \\ 0.5000 & {- 0.5000} & {0. - {0.5000{\mathbb{i}}}} \\ 0.5000 & 0.5000 & {- 0.5000} \\ {- 0.5000} & 0.5000 & {0 - {0.5000{\mathbb{i}}}} \end{matrix}$ ${V\; 3\left( {:{,{:{,54}}}} \right)} = \begin{matrix} {{- 0.0891} - {0.1737{\mathbb{i}}}} & {{- 0.5874} + {0.0189{\mathbb{i}}}} & {0.4181 + {0.5480{\mathbb{i}}}} \\ {{- 0.2160} + {0.0403{\mathbb{i}}}} & {{- 04608} - {0.5955{\mathbb{i}}}} & {0.2155 - {0.3594{\mathbb{i}}}} \\ {0.3530 + {0.5446{\mathbb{i}}}} & {{- 0.2058} - {0.0338{\mathbb{i}}}} & {{- 0.2629} + {0.1610{\mathbb{i}}}} \\ {0.1657 + {0.6818{\mathbb{i}}}} & {0.1876 - {0.0948{\mathbb{i}}}} & {0.4723 + {0.1766{\mathbb{i}}}} \end{matrix}$ ${V\; 3\left( {:{,{:{,55}}}} \right)} = \begin{matrix} {0.3162 - {0.1330{\mathbb{i}}}} & {{- 0.5287} - {0.1679{\mathbb{i}}}} & {0.1564 + {0.5043{\mathbb{i}}}} \\ {{- 0.2451} + {0.4651{\mathbb{i}}}} & {0.2831 + {0.1482{\mathbb{i}}}} & {0.4827 + {0.6000{\mathbb{i}}}} \\ {0.1535 - {0.5047{\mathbb{i}}}} & {0.7325 - {0.1862{\mathbb{i}}}} & {0.0058 + {0.2362{\mathbb{i}}}} \\ {{- 0.4972} - {0.2836{\mathbb{i}}}} & {0.0497 + {0.1282{\mathbb{i}}}} & {0.1692 - {0.2094{\mathbb{i}}}} \end{matrix}$ ${V\; 3\left( {:{,{:{,56}}}} \right)} = \begin{matrix} {0.1157 + {0.3948{\mathbb{i}}}} & {0.5787 - {0.3292{\mathbb{i}}}} & {0.2943 - {0.1843{\mathbb{i}}}} \\ {0.2814 - {0.3175{\mathbb{i}}}} & {0.2525 - {0.2620{\mathbb{i}}}} & {{- 0.5641} + {0.4338{\mathbb{i}}}} \\ {0.4393 + {0.0781{\mathbb{i}}}} & {0.3512 + {0.0363{\mathbb{i}}}} & {0.2819 + {0.4758{\mathbb{i}}}} \\ {0.3704 - {0.5609{\mathbb{i}}}} & {0.1336 + {0.5309{\mathbb{i}}}} & {0.1521 - {0.2100{\mathbb{i}}}} \end{matrix}$

${V\; 3\left( {:{,{:{,57}}}} \right)} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 \\ {0.3536 + {0.3536{\mathbb{i}}}} & {{- 0.3536} + {0.3536{\mathbb{i}}}} & {0.3536 - {0.3536{\mathbb{i}}}} \\ {0 + {0.5000{\mathbb{i}}}} & {0 - {0.5000{\mathbb{i}}}} & {0 - {0.5000{\mathbb{i}}}} \\ {{- 0.3536} + {0.3536{\mathbb{i}}}} & {0.3536 + {0.3536{\mathbb{i}}}} & {{- 0.3536} - {0.3536{\mathbb{i}}}} \end{matrix}$ ${V\; 3\left( {:{,{:{,58}}}} \right)} = \begin{matrix} {{- 0.6183} - {0.3893{\mathbb{i}}}} & {{- 0.3104} - {0.0769{\mathbb{i}}}} & {0.2262 - {0.1301{\mathbb{i}}}} \\ {0.0439 - {0.4301{\mathbb{i}}}} & {0.0424 - {0.0174{\mathbb{i}}}} & {0.4717 - {0.0872{\mathbb{i}}}} \\ {{- 0.3231} + {0.0445{\mathbb{i}}}} & {{- 0.0056} + {0.4980{\mathbb{i}}}} & {0.2818 + {0.6867{\mathbb{i}}}} \\ {{- 0.2461} - {0.3351{\mathbb{i}}}} & {0.7828 + {0.1865{\mathbb{i}}}} & {{- 0.3883} + {0.0121{\mathbb{i}}}} \end{matrix}$ ${V\; 3\left( {:{,{:{,59}}}} \right)} = \begin{matrix} {0.3566 - {0.2418{\mathbb{i}}}} & {0.0169 - {0.4765{\mathbb{i}}}} & {{- 0.5447} - {0.3859{\mathbb{i}}}} \\ {{- 0.1924} + {0.0246{\mathbb{i}}}} & {{- 0.3480} + {0.4512{\mathbb{i}}}} & {0.0113 - {0.6545{\mathbb{i}}}} \\ {{- 0.5527} + {0.1626{\mathbb{i}}}} & {{- 0.3128} - {0.0721{\mathbb{i}}}} & {{- 0.3061} - {0.1765{\mathbb{i}}}} \\ {{- 0.4030} - {0.5315{\mathbb{i}}}} & {0.5097 + {0.2917{\mathbb{i}}}} & {0.0150 + {0.0288{\mathbb{i}}}} \end{matrix}$ ${V\; 3\left( {:{,{:{,60}}}} \right)} = \begin{matrix} {0.0072 + {0.6080{\mathbb{i}}}} & {0.0164 + {0.4489{\mathbb{i}}}} & {0.2019 + {0.3032{\mathbb{i}}}} \\ {{- 0.1531} - {0.34051{\mathbb{i}}}} & {{- 0.1388} + {0.8192{\mathbb{i}}}} & {0.0946 - {0.3643{\mathbb{i}}}} \\ {0.4820 + {0.4643{\mathbb{i}}}} & {{- 0.2314} + {0.1231{\mathbb{i}}}} & {{- 0.4842} - {0.3928{\mathbb{i}}}} \\ {{- 0.1738} - {0.1132{\mathbb{i}}}} & {{- 0.1247} - {0.1538{\mathbb{i}}}} & {{- 0.4937} - {0.3051{\mathbb{i}}}} \end{matrix}$

${V\; 3\left( {:{,{:{,61}}}} \right)} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 \\ {{- 0.3536} + {0.3536{\mathbb{i}}}} & {{- 0.3536} - {0.3536{\mathbb{i}}}} & {0.3536 - {0.3536{\mathbb{i}}}} \\ {0 - {0.5000{\mathbb{i}}}} & {0 + {0.5000{\mathbb{i}}}} & {0 - {0.5000{\mathbb{i}}}} \\ {0.3536 + {0.3536{\mathbb{i}}}} & {0.3536 - {0.3536{\mathbb{i}}}} & {{- 0.3536} - {0.3536{\mathbb{i}}}} \end{matrix}$ ${V\; 3\left( {:{,{:{,62}}}} \right)} = \begin{matrix} {{- 0.4387} - {0.0343{\mathbb{i}}}} & {0.5100 - {0.0764{\mathbb{i}}}} & {0.2952 + {0.4317{\mathbb{i}}}} \\ {0.5062 - {0.5972{\mathbb{i}}}} & {0.2808 - {0.2160{\mathbb{i}}}} & {0.1570 + {0.2361{\mathbb{i}}}} \\ {{- 0.2611} + {0.2853{\mathbb{i}}}} & {{- 0.1195} + {0.0730{\mathbb{i}}}} & {{- 0.1784} + {0.6546{\mathbb{i}}}} \\ {{- 0.0591} + {0.2011{\mathbb{i}}}} & {0.7066 - {0.2995{\mathbb{i}}}} & {{- 0.3568} - {0.2417{\mathbb{i}}}} \end{matrix}$ ${V\; 3\left( {:{,{:{,63}}}} \right)} = \begin{matrix} {{- 0.0757} - {0.5813{\mathbb{i}}}} & {0.1457 - {0.0933{\mathbb{i}}}} & {{- 0.5106} + {0.2643{\mathbb{i}}}} \\ {0.1019 + {0.2454{\mathbb{i}}}} & {0.3241 + {0.2186{\mathbb{i}}}} & {0.2092 - {0.3897{\mathbb{i}}}} \\ {0.5989 + {0.2606{\mathbb{i}}}} & {{- 0.5728} - {0.0137{\mathbb{i}}}} & {{- 0.1198} + {0.3689{\mathbb{i}}}} \\ {{- 0.3284} + {0.2264{\mathbb{i}}}} & {{- 0.1976} + {0.0678{\mathbb{i}}}} & {{- 0.5620} - {0.0868{\mathbb{i}}}} \end{matrix}$ ${V\; 3\left( {:{,{:{,64}}}} \right)} = \begin{matrix} {0.2711 - {0.7070{\mathbb{i}}}} & {{- 0.2911} + {0.2527{\mathbb{i}}}} & {{- 0.1768} + {0.3248{\mathbb{i}}}} \\ {{- 0.4123} - {0.1234{\mathbb{i}}}} & {0.0110 - {0.0641{\mathbb{i}}}} & {{- 0.4612} - {0.6234{\mathbb{i}}}} \\ {0.4714 - {0.0224{\mathbb{i}}}} & {0.3098 + {0.2417{\mathbb{i}}}} & {0.0189 - {0.4271{\mathbb{i}}}} \\ {{- 0.0617} - {0.1224{\mathbb{i}}}} & {{- 0.4468} - {0.7023{\mathbb{i}}}} & {0.2720 - {0.0718{\mathbb{i}}}} \end{matrix}$

-   -   Final Rank 4 Codebook:

${V\; 4\left( {:{,{:{,1}}}} \right)} = \begin{matrix} 0.5000 & 0.5000 & {- 0.5000} & {- 0.5000} \\ 0.5000 & {- 0.5000} & {- 0.5000} & 0.5000 \\ 0.5000 & 0.5000 & 0.5000 & 0.5000 \\ 0.5000 & {- 0.5000} & 0.5000 & {- 0.5000} \end{matrix}$ ${V\; 4\left( {:{,{:{,2}}}} \right)} = \begin{matrix} 0.5000 & 0.5000 & {- 0.5000} & {- 0.5000} \\ {0 + {0.5000{\mathbb{i}}}} & {0 - {0.5000{\mathbb{i}}}} & {0 - {0.5000{\mathbb{i}}}} & {0 + {0.5000{\mathbb{i}}}} \\ 0.5000 & 0.5000 & 0.5000 & 0.5000 \\ {0 + {0.5000{\mathbb{i}}}} & {0 - {0.5000{\mathbb{i}}}} & {0 + {0.5000{\mathbb{i}}}} & {0 - {0.5000{\mathbb{i}}}} \end{matrix}$ ${V\; 4\left( {:{,{:{,3}}}} \right)} = \begin{matrix} 0.5000 & 0.5000 & {- 0.5000} & {- 0.5000} \\ 0.5000 & {- 0.5000} & {0 - {0.5000{\mathbb{i}}}} & {0 + {0.5000{\mathbb{i}}}} \\ 0.5000 & 0.5000 & 0.5000 & 0.5000 \\ 0.5000 & {- 0.5000} & {0 + {0.5000{\mathbb{i}}}} & {0 - {0.5000{\mathbb{i}}}} \end{matrix}$ ${V\; 4\left( {:{,{:{,4}}}} \right)} = \begin{matrix} 0.5000 & 0.5000 & {- 0.5000} & {- 0.5000} \\ {0 + {0.5000{\mathbb{i}}}} & {0 - {0.5000{\mathbb{i}}}} & {- 0.5000} & 0.5000 \\ 0.5000 & 0.5000 & 0.5000 & 0.5000 \\ {0 + {0.5000{\mathbb{i}}}} & {0 - {0.5000{\mathbb{i}}}} & 0.5000 & {- 0.5000} \end{matrix}$

${V\; 4\left( {\text{:},\text{:},5} \right)} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 & 0.5000 \\ 0.5000 & {0 + {0.5000{\mathbb{i}}}} & {- 0.5000} & {0 - {0.5000{\mathbb{i}}}} \\ 0.5000 & {- 0.5000} & 0.5000 & {- 0.5000} \\ {- 0.5000} & {0 + {0.5000{\mathbb{i}}}} & 0.5000 & {0 - {0.5000{\mathbb{i}}}} \end{matrix}$ ${V\; 4\left( {\text{:},\text{:},6} \right)} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 & 0.5000 \\ {0.3536 + {0.3536{\mathbb{i}}}} & {{- 0.3536} + {0.3536{\mathbb{i}}}} & {{- 0.3536} - {0.3536{\mathbb{i}}}} & {0.3536 - {0.3536{\mathbb{i}}}} \\ {0 + {0.5000{\mathbb{i}}}} & {0 - {0.5000{\mathbb{i}}}} & {0 + {0.5000{\mathbb{i}}}} & {0 - {0.5000{\mathbb{i}}}} \\ {{- 0.3536} + {0.3536{\mathbb{i}}}} & {0.3536 + {0.3536{\mathbb{i}}}} & {0.3536 - {0.3536{\mathbb{i}}}} & {{- 0.3536} - {0.3536{\mathbb{i}}}} \end{matrix}$ ${{V4}\left( {\text{:},\text{:},7} \right)} = \begin{matrix} {0.3260 + {0.6774{\mathbb{i}}}} & {0.4688 + {0.2170{\mathbb{i}}}} & {0.3008 + {0.0981{\mathbb{i}}}} & {0.1395 - {0.2201{\mathbb{i}}}} \\ {0.3254 + {0.1709{\mathbb{i}}}} & {{- 0.2948} - {0.0700{\mathbb{i}}}} & {{- 0.1167} + {0.7005{\mathbb{i}}}} & {{- 0.4967} + {0.1489{\mathbb{i}}}} \\ {0.3254 + {0.1709{\mathbb{i}}}} & {{- 0.4598} - {0.2965{\mathbb{i}}}} & {{- 0.1297} + {0.0716{\mathbb{i}}}} & {0.7351 + {0.0569{\mathbb{i}}}} \\ {{- 0.0250} + {0.4051{\mathbb{i}}}} & {{- 0.5846} - {0.0154{\mathbb{i}}}} & {0.3317 - {0.5135{\mathbb{i}}}} & {{- 0.3430} - {0.0437{\mathbb{i}}}} \end{matrix}$ ${{V4}\left( {\text{:},\text{:},8} \right)} = \begin{matrix} {0.1499 + {0.0347{\mathbb{i}}}} & {{- 0.4377} + {0.4498{\mathbb{i}}}} & {0.2079 + {0.3012{\mathbb{i}}}} & {{- 0.4172} - {0.5239{\mathbb{i}}}} \\ {0.5009 + {0.3071{\mathbb{i}}}} & {{- 0.1173} + {0.2079{\mathbb{i}}}} & {0.4297 + {0.2345{\mathbb{i}}}} & {0.4687 + {0.3722{\mathbb{i}}}} \\ {0.1505 + {0.5412{\mathbb{i}}}} & {{- 0.2098} + {0.2390{\mathbb{i}}}} & {{- 0.2835} - {0.7076{\mathbb{i}}}} & {{- 0.0338} - {0.0330{\mathbb{i}}}} \\ {0.1505 + {0.5412{\mathbb{i}}}} & {0.6133 - {0.2682{\mathbb{i}}}} & {0.1669 + {0.1324{\mathbb{i}}}} & {{- 0.1311} - {0.4169{\mathbb{i}}}} \end{matrix}$

${{V4}\left( {\text{:},\text{:},9} \right)} = \begin{matrix} {0.0918 + {0.3270`{\mathbb{i}}}} & {0.4585 - {0.3521{\mathbb{i}}}} & {0.4797 - {0.5371{\mathbb{i}}}} & {0.1760 - {0.0285{\mathbb{i}}}} \\ {0.1311 + {0.6387{\mathbb{i}}}} & {{- 0.5369} + {0.2492{\mathbb{i}}}} & {0.2256 + {0.0214{\mathbb{i}}}} & {0.0252 - {0.4154{\mathbb{i}}}} \\ {0.2473 + {0.0541{\mathbb{i}}}} & {{- 0.1391} + {0.3154{\mathbb{i}}}} & {{- 0.1147} - {0.1411{\mathbb{i}}}} & {0.7276 + {0.5046{\mathbb{i}}}} \\ {0.4815 + {0.4045{\mathbb{i}}}} & {0.2872 - {0.3378{\mathbb{i}}}} & {{- 0.2334} + {0.5852{\mathbb{i}}}} & {0.0121 + {0.1041{\mathbb{i}}}} \end{matrix}$ ${{V4}\left( {\text{:},\text{:},10} \right)} = \begin{matrix} {0.0918 + {0.3270`{\mathbb{i}}}} & {{- 0.7326} - {0.3303{\mathbb{i}}}} & {0.1180 - {0.0570{\mathbb{i}}}} & {0.3116 - {0.3531{\mathbb{i}}}} \\ {{- 0.1311} - {0.6387{\mathbb{i}}}} & {{- 0.1521} + {0.1245{\mathbb{i}}}} & {{- 0.3996} - {0.1865{\mathbb{i}}}} & {0.5845 - {0.0141{\mathbb{i}}}} \\ {0.2473 + {0.0541{\mathbb{i}}}} & {0.0048 - {0.3893{\mathbb{i}}}} & {{- 0.8009} - {0.0266{\mathbb{i}}}} & {{- 0.3599} - {0.1123{\mathbb{i}}}} \\ {{- 0.4815} - {0.4045{\mathbb{i}}}} & {{- 0.3093} - {0.2615{\mathbb{i}}}} & {0.1996 - {0.3262{\mathbb{i}}}} & {{- 0.5417} + {0.0279{\mathbb{i}}}} \end{matrix}$ ${{V4}\left( {\text{:},\text{:},11} \right)} = \begin{matrix} {0.3841 + {0.3851{\mathbb{i}}}} & {{- 0.4259} - {0.6865{\mathbb{i}}}} & {0.1888 - {0.1148{\mathbb{i}}}} & {0.0505 - {0.0112{\mathbb{i}}}} \\ {0.0056 - {0.3076{\mathbb{i}}}} & {0.0000 - {0.0163{\mathbb{i}}}} & {0.5849 + {0.0396{\mathbb{i}}}} & {0.1477 + {0.7346{\mathbb{i}}}} \\ {0.2285 + {0.6580{\mathbb{i}}}} & {{- 0.0029} + {0.5625{\mathbb{i}}}} & {0.2943 + {0.2161{\mathbb{i}}}} & {0.2547 - {0.0108{\mathbb{i}}}} \\ {{- 0.3448} - {0.0735{\mathbb{i}}}} & {{- 0.1664} + {0.0538{\mathbb{i}}}} & {0.6828 - {0.0892{\mathbb{i}}}} & {{- 0.3558} - {0.4943{\mathbb{i}}}} \end{matrix}$ ${{V4}\left( {\text{:},\text{:},12} \right)} = \begin{matrix} {0.3260 + {0.6774{\mathbb{i}}}} & {{- 0.2522} + {0.2830{\mathbb{i}}}} & {0.0842 - {0.4310{\mathbb{i}}}} & {0.0964 + {0.2984{\mathbb{i}}}} \\ {{- 0.3254} - {0.1709{\mathbb{i}}}} & {{- 0.2511} + {0.3562{\mathbb{i}}}} & {{- 0.5001} - {0.4577{\mathbb{i}}}} & {{- 0.4236} - {0.1898{\mathbb{i}}}} \\ {0.3254 + {0.1709{\mathbb{i}}}} & {{- 0.6354} - {0.2810{\mathbb{i}}}} & {{- 0.3335} + {0.4366{\mathbb{i}}}} & {{- 0.1484} - {0.2414{\mathbb{i}}}} \\ {0.0250 - {0.4051{\mathbb{i}}}} & {{- 0.2798} - {0.3246{\mathbb{i}}}} & {{- 0.1512} - {0.1513{\mathbb{i}}}} & {0.0306 + {0.7778{\mathbb{i}}}} \end{matrix}$

${{V4}\left( {\text{:},\text{:},13} \right)} = \begin{matrix} {{- 0.0918} - {0.3270{\mathbb{i}}}} & {0.4517 - {0.1172{\mathbb{i}}}} & {{- 0.0439} + {0.5591{\mathbb{i}}}} & {0.3594 - {0.4725{\mathbb{i}}}} \\ {{- 0.2473} - {0.0541{\mathbb{i}}}} & {{- 0.3568} + {0.3992{\mathbb{i}}}} & {{- 0.4696} - {0.0932{\mathbb{i}}}} & {{- 0.2384} - {0.6026{\mathbb{i}}}} \\ {0.1311 + {0.6387{\mathbb{i}}}} & {0.0918 + {0.4242{\mathbb{i}}}} & {0.4085 - {0.0481{\mathbb{i}}}} & {0.3325 - {0.3269{\mathbb{i}}}} \\ {0.4815 + {0.4045{\mathbb{i}}}} & {{- 0.3837} - {0.3999{\mathbb{i}}}} & {{- 0.3005} + {0.4437{\mathbb{i}}}} & {{- 0.0155} - {0.1000{\mathbb{i}}}} \end{matrix}$ ${{V4}\left( {\text{:},\text{:},14} \right)} = \begin{matrix} {{- 0.0337} - {0.6193`{\mathbb{i}}}} & {0.3983 + {0.5597{\mathbb{i}}}} & {{- 0.1922} - {0.0358{\mathbb{i}}}} & {{- 0.2096} + {0.2475{\mathbb{i}}}} \\ {{- 0.4621} - {0.5019{\mathbb{i}}}} & {{- 0.5583} + {0.0724{\mathbb{i}}}} & {0.2420 - {0.0149{\mathbb{i}}}} & {0.3931 - {0.0653{\mathbb{i}}}} \\ {0.2280 + {0.1515{\mathbb{i}}}} & {0.2543 + {0.2747{\mathbb{i}}}} & {0.7756 - {0.0378{\mathbb{i}}}} & {0.3367 + {0.2619{\mathbb{i}}}} \\ {0.2280 + {0.1515{\mathbb{i}}}} & {{- 0.0069} + {0.2664{\mathbb{i}}}} & {{- 0.5242} + {0.1589{\mathbb{i}}}} & {0.7417 + {0.0629{\mathbb{i}}}} \end{matrix}$ ${{V4}\left( {\text{:},\text{:},15} \right)} = \begin{matrix} {{- 0.4422} - {0.0928{\mathbb{i}}}} & {0.3609 + {0.0743{\mathbb{i}}}} & {{- 0.6309} - {0.1183{\mathbb{i}}}} & {0.2159 - {0.4488{\mathbb{i}}}} \\ {{- 0.2479} - {0.5606{\mathbb{i}}}} & {{- 0.0447} - {0.2624{\mathbb{i}}}} & {{- 0.3156} + {0.2879{\mathbb{i}}}} & {{- 0.0988} + {0.6010{\mathbb{i}}}} \\ {0.2479 + {0.5606{\mathbb{i}}}} & {0.2646 - {0.2157{\mathbb{i}}}} & {{- 0.3493} - {0.1821{\mathbb{i}}}} & {0.3113 + {0.5056{\mathbb{i}}}} \\ {0.0137 + {0.2102{\mathbb{i}}}} & {{- 0.7438} + {0.3516{\mathbb{i}}}} & {{- 0.4975} - {0.0539{\mathbb{i}}}} & {{- 0.1643} + {0.0375{\mathbb{i}}}} \end{matrix}$ ${{V4}\left( {\text{:},\text{:},16} \right)} = \begin{matrix} {{- 0.4422} - {0.0928{\mathbb{i}}}} & {{- 0.4197} + {0.4817{\mathbb{i}}}} & {0.0820 + {0.2932{\mathbb{i}}}} & {0.5394 + {0.0634{\mathbb{i}}}} \\ {0.2479 + {0.5606{\mathbb{i}}}} & {{- 0.2858} - {0.0562{\mathbb{i}}}} & {0.0717 - {0.5527{\mathbb{i}}}} & {0.3834 + {0.2860{\mathbb{i}}}} \\ {0.2479 + {0.5606{\mathbb{i}}}} & {{- 0.0712} + {0.2154{\mathbb{i}}}} & {{- 0.2709} + {0.6053{\mathbb{i}}}} & {{- 0.2652} + {0.2505{\mathbb{i}}}} \\ {{- 0.0137} - {0.2102{\mathbb{i}}}} & {0.1024 + {0.6671{\mathbb{i}}}} & {{- 0.0386} - {0.3943{\mathbb{i}}}} & {{- 0.3941} + {0.4333{\mathbb{i}}}} \end{matrix}$

${{V4}\left( {\text{:},\text{:},17} \right)} = \begin{matrix} {{- 0.3841} - {0.3851{\mathbb{i}}}} & {0.3849 + {0.1986`{\mathbb{i}}}} & {{- 0.5420} - {0.2820{\mathbb{i}}}} & {0.3395 + {0.1672{\mathbb{i}}}} \\ {{- 0.0056} + {0.3076{\mathbb{i}}}} & {{- 0.1824} - {0.0094{\mathbb{i}}}} & {0.1989 + {0.1166{\mathbb{i}}}} & {0.6694 + {0.6089{\mathbb{i}}}} \\ {0.3448 + {0.0735{\mathbb{i}}}} & {0.2254 + {0.6795{\mathbb{i}}}} & {{- 0.1979} + {0.5597{\mathbb{i}}}} & {{- 0.0690} + {0.0775{\mathbb{i}}}} \\ {{- 0.2285} - {0.6580{\mathbb{i}}}} & {{- 0.5154} + {0.0292{\mathbb{i}}}} & {0.0915 + {0.4612{\mathbb{i}}}} & {0.1367 - {0.0921{\mathbb{i}}}} \end{matrix}$ ${{V4}\left( {\text{:},\text{:},18} \right)} = \begin{matrix} {{- 0.0918} - {0.3270{\mathbb{i}}}} & {0.6638 - {0.4917{\mathbb{i}}}} & {{- 0.0134} - {0.1109{\mathbb{i}}}} & {0.3780 + {0.2166{\mathbb{i}}}} \\ {0.2473 + {0.0541{\mathbb{i}}}} & {{- 0.1566} - {0.1262{\mathbb{i}}}} & {0.0722 - {0.9308{\mathbb{i}}}} & {{- 0.1132} + {0.1055{\mathbb{i}}}} \\ {0.1311 + {0.6387{\mathbb{i}}}} & {{- 0.0029} - {0.3366{\mathbb{i}}}} & {{- 0.1758} + {0.1916{\mathbb{i}}}} & {{- 0.1479} + {0.6100{\mathbb{i}}}} \\ {{- 0.4815} - {0.4045{\mathbb{i}}}} & {{- 0.3978} + {0.0753{\mathbb{i}}}} & {0.2200 - {0.0040{\mathbb{i}}}} & {{- 0.0224} + {0.6259{\mathbb{i}}}} \end{matrix}$ ${{V4}\left( {\text{:},\text{:},19} \right)} = \begin{matrix} {0.3841 + {0.3851{\mathbb{i}}}} & {0.2420 - {0.2055{\mathbb{i}}}} & {{- 0.1388} + {0.6904{\mathbb{i}}}} & {0.3273 + {0.0199{\mathbb{i}}}} \\ {{- 0.3076} - {0.0056{\mathbb{i}}}} & {{- 0.5655} - {0.6754{\mathbb{i}}}} & {{- 0.2104} + {0.2029{\mathbb{i}}}} & {{- 0.1634} + {0.1312{\mathbb{i}}}} \\ {0.3448 + {0.0735{\mathbb{i}}}} & {0.0981 + {0.0467{\mathbb{i}}}} & {0.1060 + {0.1310{\mathbb{i}}}} & {{- 0.7933} + {0.4542{\mathbb{i}}}} \\ {{- 0.6580} + {0.2285{\mathbb{i}}}} & {0.3298 - {0.0517{\mathbb{i}}}} & {0.5748 + {0.2447{\mathbb{i}}}} & {{- 0.0516} + {0.1018{\mathbb{i}}}} \end{matrix}$ ${{V4}\left( {\text{:},\text{:},20} \right)} = \begin{matrix} {0.0918 + {0.3270{\mathbb{i}}}} & {{- 0.2841} - {0.7969{\mathbb{i}}}} & {0.3713 - {0.0745{\mathbb{i}}}} & {0.0639 + {0.1460{\mathbb{i}}}} \\ {{- 0.0541} + {0.2473{\mathbb{i}}}} & {0.0227 + {0.2058`{\mathbb{i}}}} & {0.5630 - {0.0230{\mathbb{i}}}} & {{- 0.0999} - {0.7520\;{\mathbb{i}}}} \\ {0.1311 + {0.6387{\mathbb{i}}}} & {0.2505 + {0.3816{\mathbb{i}}}} & {0.1527 - {0.2390{\mathbb{i}}}} & {0.3715 + {0.3849{\mathbb{i}}}} \\ {{- 0.4045} + {0.4815{\mathbb{i}}}} & {{- 0.0898} - {0.1576`{\mathbb{i}}}} & {{- 0.6437} - {0.2105{\mathbb{i}}}} & {0.0144 - {0.3358{\mathbb{i}}}} \end{matrix}$

${{V4}\left( {\text{:},\text{:},21} \right)} = \begin{matrix} {0.0337 + {0.6193{\mathbb{i}}}} & {0.3344 - {0.1640{\mathbb{i}}}} & {{- 0.2485} + {0.4791{\mathbb{i}}}} & {0.4000 + {0.1590{\mathbb{i}}}} \\ {{- 0.5019} + {0.4621{\mathbb{i}}}} & {{- 0.1475} + {0.2641{\mathbb{i}}}} & {0.5893 + {0.1330{\mathbb{i}}}} & {{- 0.1383} - {0.2427`{\mathbb{i}}}} \\ {0.2280 + {0.1515{\mathbb{i}}}} & {{- 0.1383} + {0.3182{\mathbb{i}}}} & {{- 0.4811} + {0.3240{\mathbb{i}}}} & {{- 0.5161} - {0.4493{\mathbb{i}}}} \\ {{- 0.1515} + {0.2280{\mathbb{i}}}} & {{- 0.3105} - {0.7437{\mathbb{i}}}} & {{- 0.0811} - {0.0251{\mathbb{i}}}} & {{- 0.4817} + {0.1909{\mathbb{i}}}} \end{matrix}$ ${{V4}\left( {\text{:},\text{:},22} \right)} = \begin{matrix} {0.0337 + {0.6193{\mathbb{i}}}} & {0.4332 + {0.1612{\mathbb{i}}}} & {{- 0.4077} - {0.0486{\mathbb{i}}}} & {0.4750 + {0.0865{\mathbb{i}}}} \\ {0.5019 - {0.4621{\mathbb{i}}}} & {{- 0.2499} + {0.4647{\mathbb{i}}}} & {{- 0.3727} - {0.1094{\mathbb{i}}}} & {0.2747 + {0.1727{\mathbb{i}}}} \\ {0.2280 + {0.1515{\mathbb{i}}}} & {0.1294 + {0.5446{\mathbb{i}}}} & {0.6827 - {0.1403{\mathbb{i}}}} & {0.1161 - {0.3354{\mathbb{i}}}} \\ {0.1515 - {0.2280{\mathbb{i}}}} & {0.3535 - {0.2640{\mathbb{i}}}} & {0.4328 + {0.0863{\mathbb{i}}}} & {0.3135 + {0.6613{\mathbb{i}}}} \end{matrix}$ ${{V4}\left( {\text{:},\text{:},23} \right)} = \begin{matrix} {0.4422 + {0.0928{\mathbb{i}}}} & {0.5948 - {0.2086{\mathbb{i}}}} & {{- 0.6155} + {0.0029{\mathbb{i}}}} & {0.0246 + {0.1384{\mathbb{i}}}} \\ {0.5606 - {0.2479{\mathbb{i}}}} & {{- 0.6101} + {0.1811{\mathbb{i}}}} & {{- 0.2226} - {0.2992{\mathbb{i}}}} & {0.0747 + {0.2731{\mathbb{i}}}} \\ {0.2479 + {0.5606{\mathbb{i}}}} & {{- 0.1244} + {0.3625{\mathbb{i}}}} & {0.0659 + {0.6219{\mathbb{i}}}} & {0.2542 + {0.1473{\mathbb{i}}}} \\ {0.2102 - {0.0137{\mathbb{i}}}} & {{- 0.1261} + {0.1870{\mathbb{i}}}} & {{- 0.1866} + {0.2370{\mathbb{i}}}} & {{- 0.7184} - {0.5456{\mathbb{i}}}} \end{matrix}$ ${{V4}\left( {\text{:},\text{:},24} \right)} = \begin{matrix} {0.3841 + {0.3851{\mathbb{i}}}} & {{- 0.7276} + {0.3231{\mathbb{i}}}} & {{- 0.0746} + {0.0002{\mathbb{i}}}} & {0.1968 + {0.1617{\mathbb{i}}}} \\ {0.3076 + {0.0056{\mathbb{i}}}} & {0.0623 - {0.3109{\mathbb{i}}}} & {0.7114 + {0.3826{\mathbb{i}}}} & {0.3902 + {0.0102{\mathbb{i}}}} \\ {0.3448 + {0.0735{\mathbb{i}}}} & {{- 0.0316} - {0.3607{\mathbb{i}}}} & {{- 0.4356} + {0.1774{\mathbb{i}}}} & {0.0838 - {0.7186{\mathbb{i}}}} \\ {0.6580 - {0.2285{\mathbb{i}}}} & {0.1496 - {0.3350{\mathbb{i}}}} & {{- 0.1859} - {0.2936{\mathbb{i}}}} & {{- 0.1918} + {0.4718{\mathbb{i}}}} \end{matrix}$

${{V4}\left( {\text{:},\text{:},25} \right)} = \begin{matrix} {{- 0.3841} - {0.3851{\mathbb{i}}}} & {0.2650 + {0.1831{\mathbb{i}}}} & {0.2046 + {0.4515{\mathbb{i}}}} & {0.2200 + {0.5535{\mathbb{i}}}} \\ {0.3076 + {0.0056{\mathbb{i}}}} & {0.0741 + {0.1142{\mathbb{i}}}} & {{- 0.2010} + {0.6413{\mathbb{i}}}} & {0.4491 - {0.4832{\mathbb{i}}}} \\ {0.2285 + {0.6580{\mathbb{i}}}} & {0.3746 + {0.4660{\mathbb{i}}}} & {0.2087 + {0.1370{\mathbb{i}}}} & {{- 0.2423} + {0.1904{\mathbb{i}}}} \\ {{- 0.0735} + {0.3448{\mathbb{i}}}} & {{- 0.0601} - {0.7188{\mathbb{i}}}} & {{- 0.1085} + {0.4782{\mathbb{i}}}} & {{- 0.2656} + {0.2110{\mathbb{i}}}} \end{matrix}$ ${{V4}\left( {\text{:},\text{:},26} \right)} = \begin{matrix} {{- 0.3260} - {0.6774{\mathbb{i}}}} & {0.4170 + {0.2673{\mathbb{i}}}} & {0.0729 - {0.3667{\mathbb{i}}}} & {{- 0.1258} + {0.1841{\mathbb{i}}}} \\ {0.1709 - {0.3254{\mathbb{i}}}} & {{- 0.0605} + {0.0504{\mathbb{i}}}} & {{- 0.2440} + {0.1836{\mathbb{i}}}} & {0.8489 + {0.2116{\mathbb{i}}}} \\ {0.3254 + {0.1709{\mathbb{i}}}} & {0.3145 + {0.3209{\mathbb{i}}}} & {{- 0.6599} - {0.3142{\mathbb{i}}}} & {{- 0.0292} - {0.3579{\mathbb{i}}}} \\ {{- 0.4051} - {0.0250{\mathbb{i}}}} & {{- 0.4946} + {0.5495{\mathbb{i}}}} & {{- 0.3770} + {0.3011{\mathbb{i}}}} & {{- 0.2007} + {0.1251{\mathbb{i}}}} \end{matrix}$ ${{V4}\left( {\text{:},\text{:},27} \right)} = \begin{matrix} {{- 0.1499} - {0.0347{\mathbb{i}}}} & {{- 0.1568} + {0.5108{\mathbb{i}}}} & {{- 0.6900} + {0.1066{\mathbb{i}}}} & {0.4419 + {0.0901{\mathbb{i}}}} \\ {0.3071 - {0.5009{\mathbb{i}}}} & {{- 0.2476} + {0.2774{\mathbb{i}}}} & {0.3823 - {0.1521{\mathbb{i}}}} & {0.3815 - {0.4492{\mathbb{i}}}} \\ {0.1505 + {0.5412{\mathbb{i}}}} & {0.1736 - {0.2279{\mathbb{i}}}} & {{- 0.1974} - {0.4916{\mathbb{i}}}} & {0.3241 - {0.4623{\mathbb{i}}}} \\ {{- 0.5412} + {0.1505{\mathbb{i}}}} & {{- 0.6443} - {0.2811{\mathbb{i}}}} & {0.0584 + {0.2373{\mathbb{i}}}} & {0.0318 - {0.3599{\mathbb{i}}}} \end{matrix}$ ${{V4}\left( {\text{:},\text{:},28} \right)} = \begin{matrix} {{- 0.1499} - {0.0347{\mathbb{i}}}} & {{- 0.4915} + {0.5944{\mathbb{i}}}} & {0.0644 - {0.0145{\mathbb{i}}}} & {{- 0.1689} - {0.5903{\mathbb{i}}}} \\ {{- 0.3071} + {0.5009{\mathbb{i}}}} & {{- 0.1046} - {0.3964{\mathbb{i}}}} & {{- 0.5642} + {0.0031{\mathbb{i}}}} & {{- 0.3416} - {0.2274{\mathbb{i}}}} \\ {0.1505 + {0.5412{\mathbb{i}}}} & {0.2288 + {0.3710{\mathbb{i}}}} & {{- 0.2561} - {0.0135{\mathbb{i}}}} & {0.6471 - {0.0998{\mathbb{i}}}} \\ {0.5412 - {0.1505{\mathbb{i}}}} & {{- 0.2089} + {0.0583{\mathbb{i}}}} & {{- 0.4070} - {0.6677{\mathbb{i}}}} & {{- 0.1176} + {0.1097{\mathbb{i}}}} \end{matrix}$

${{V4}\left( {\text{:},\text{:},29} \right)} = \begin{matrix} {{- 0.0918} - {0.3270{\mathbb{i}}}} & {0.0849 - {0.1823{\mathbb{i}}}} & {0.5488 + {0.2367{\mathbb{i}}}} & {{- 0.3534} - {0.6018{\mathbb{i}}}} \\ {{- 0.6387} + {0.1311{\mathbb{i}}}} & {{- 0.1210} - {0.0233{\mathbb{i}}}} & {{- 0.2347} - {0.3498{\mathbb{i}}}} & {0.3248 - {0.5261{\mathbb{i}}}} \\ {0.2473 + {0.0541{\mathbb{i}}}} & {0.4127 - {0.7969{\mathbb{i}}}} & {{- 0.2180} - {0.2794{\mathbb{i}}}} & {{- 0.0107} - {0.0700{\mathbb{i}}}} \\ {0.4045 - {0.4815{\mathbb{i}}}} & {{- 0.0136} + {0.3727{\mathbb{i}}}} & {{- 0.3369} - {0.4757{\mathbb{i}}}} & {{- 0.2087} - {0.2866{\mathbb{i}}}} \end{matrix}$ ${{V4}\left( {\text{:},\text{:},30} \right)} = \begin{matrix} {{- 0.3841} - {0.3851{\mathbb{i}}}} & {0.4828 - {0.2545{\mathbb{i}}}} & {0.2119 + {0.3871{\mathbb{i}}}} & {{- 0.3824} - {0.2556{\mathbb{i}}}} \\ {{- 0.3076} - {0.0056{\mathbb{i}}}} & {{- 0.1939} + {0.1069{\mathbb{i}}}} & {{- 0.3495} + {0.6691{\mathbb{i}}}} & {0.1365 + {0.5175{\mathbb{i}}}} \\ {0.2285 + {0.6580{\mathbb{i}}}} & {0.3116 + {0.2702{\mathbb{i}}}} & {{- 0.1220} + {0.2056{\mathbb{i}}}} & {{- 0.5361} - {0.0030{\mathbb{i}}}} \\ {0.0735 - {0.3448{\mathbb{i}}}} & {{- 0.4258} + {0.5492{\mathbb{i}}}} & {0.4208 + {0.0338{\mathbb{i}}}} & {{- 0.4460} + {0.1251{\mathbb{i}}}} \end{matrix}$ ${{V4}\left( {\text{:},\text{:},31} \right)} = \begin{matrix} {0.0337 + {0.6193{\mathbb{i}}}} & {{- 0.3283} - {0.4339{\mathbb{i}}}} & {{- 0.2690} + {0.1335{\mathbb{i}}}} & {{- 0.4536} - {0.1528{\mathbb{i}}}} \\ {0.2280 + {0.1515{\mathbb{i}}}} & {{- 0.3886} - {0.2987{\mathbb{i}}}} & {0.0982 - {0.6284{\mathbb{i}}}} & {0.5255 + {0.0653{\mathbb{i}}}} \\ {0.2280 + {0.1515{\mathbb{i}}}} & {0.2457 - {0.2337{\mathbb{i}}}} & {{- 0.3618} + {0.5436{\mathbb{i}}}} & {0.6122 + {0.0946{\mathbb{i}}}} \\ {{- 0.4621} - {0.5019{\mathbb{i}}}} & {{- 0.4861} - {0.3354{\mathbb{i}}}} & {{- 0.2060} + {0.1911{\mathbb{i}}}} & {0.0216 + {0.3261{\mathbb{i}}}} \end{matrix}$ ${{V4}\left( {\text{:},\text{:},32} \right)} = \begin{matrix} {0.0918 + {0.3270{\mathbb{i}}}} & {{- 0.0348} + {0.4484{\mathbb{i}}}} & {0.4669 - {0.6204{\mathbb{i}}}} & {0.2685 + {0.0858{\mathbb{i}}}} \\ {0.4815 + {0.4045{\mathbb{i}}}} & {{- 0.4400} + {0.2694{\mathbb{i}}}} & {0.0511 + {0.4386{\mathbb{i}}}} & {{- 0.1700} + {0.3385{\mathbb{i}}}} \\ {0.1311 + {0.6387{\mathbb{i}}}} & {0.3429 - {0.0049{\mathbb{i}}}} & {0.4475 - {0.0066{\mathbb{i}}}} & {0.1480 - {0.4849{\mathbb{i}}}} \\ {{- 0.2473} - {0.0541{\mathbb{i}}}} & {0.1991 + {0.6118{\mathbb{i}}}} & {0.0419 + {0.0073{\mathbb{i}}}} & {{- 0.6988} - {0.1782{\mathbb{i}}}} \end{matrix}$

${{V4}\left( {\text{:},\text{:},33} \right)} = \begin{matrix} {0.3841 + {0.3851{\mathbb{i}}}} & {{- 0.2360} - {0.2973{\mathbb{i}}}} & {0.3316 + {0.2303{\mathbb{i}}}} & {0.0526 + {0.6279{\mathbb{i}}}} \\ {0.2285 + {0.6580{\mathbb{i}}}} & {0.0315 - {0.2366{\mathbb{i}}}} & {{- 0.3698} - {0.2862{\mathbb{i}}}} & {0.3197 - {0.3701{\mathbb{i}}}} \\ {0.3448 + {0.0735{\mathbb{i}}}} & {0.7285 + {0.0791{\mathbb{i}}}} & {0.5225 + {0.0720{\mathbb{i}}}} & {{- 0.0255} - {0.2448{\mathbb{i}}}} \\ {0.0056 - {0.3076{\mathbb{i}}}} & {0.4097 - {0.3067{\mathbb{i}}}} & {{- 0.4142} + {0.4106{\mathbb{i}}}} & {0.5048 + {0.2198{\mathbb{i}}}} \end{matrix}$ ${{V4}\left( {\text{:},\text{:},34} \right)} = \begin{matrix} {0.4422 + {0.0928{\mathbb{i}}}} & {{- 0.5245} - {0.3774{\mathbb{i}}}} & {{- 0.5623} - {0.0655{\mathbb{i}}}} & {0.2278 + {0.0771{\mathbb{i}}}} \\ {{- 0.2102} + {0.0137{\mathbb{i}}}} & {{- 0.0522} + {0.4805{\mathbb{i}}}} & {{- 0.1132} - {0.5413{\mathbb{i}}}} & {0.6451 - {0.0046{\mathbb{i}}}} \\ {{- 0.2479} - {0.5606{\mathbb{i}}}} & {0.1023 + {0.0593{\mathbb{i}}}} & {{- 0.5232} - {0.2730{\mathbb{i}}}} & {{- 0.4236} + {0.2873{\mathbb{i}}}} \\ {{- 0.5606} + {0.2479{\mathbb{i}}}} & {{- 0.5756} - {0.0591{\mathbb{i}}}} & {0.0088 - {0.1592{\mathbb{i}}}} & {{- 0.3254} - {0.3977{\mathbb{i}}}} \end{matrix}$ ${{V4}\left( {\text{:},\text{:},35} \right)} = \begin{matrix} {0.0337 + {0.6193{\mathbb{i}}}} & {0.1953 - {0.3057{\mathbb{i}}}} & {{- 0.5062} + {0.0964{\mathbb{i}}}} & {0.4500 + {0.1253{\mathbb{i}}}} \\ {{- 0.1515} + {0.2280{\mathbb{i}}}} & {{- 0.2685} - {0.1895{\mathbb{i}}}} & {0.6204 + {0.3044{\mathbb{i}}}} & {0.1623 + {0.5596{\mathbb{i}}}} \\ {{- 0.2280} - {0.1515{\mathbb{i}}}} & {{- 0.7779} - {0.3721{\mathbb{i}}}} & {{- 0.3442} - {0.2479{\mathbb{i}}}} & {{- 0.0309} + {0.0263{\mathbb{i}}}} \\ {{- 0.5019} + {0.4621{\mathbb{i}}}} & {{- 0.0216} - {0.1283{\mathbb{i}}}} & {0.2771 + {0.0109{\mathbb{i}}}} & {{- 0.1897} - {0.6361{\mathbb{i}}}} \end{matrix}$ ${{V4}\left( {\text{:},\text{:},36} \right)} = \begin{matrix} {0.0918 + {0.3270{\mathbb{i}}}} & {{- 0.6035} - {0.3342{\mathbb{i}}}} & {{- 0.3380} - {0.3706{\mathbb{i}}}} & {{- 0.0686} + {0.3905{\mathbb{i}}}} \\ {{- 0.4045} + {0.4815{\mathbb{i}}}} & {{- 0.1428} + {0.4838{\mathbb{i}}}} & {{- 0.1430} - {0.1006{\mathbb{i}}}} & {{- 0.3957} - {0.4036{\mathbb{i}}}} \\ {{- 0.1311} - {0.6387{\mathbb{i}}}} & {{- 0.1927} + {0.1626{\mathbb{i}}}} & {{- 0.0874} + {0.0678{\mathbb{i}}}} & {{- 0.6478} + {0.2818{\mathbb{i}}}} \\ {{- 0.0541} + {0.2473{\mathbb{i}}}} & {{- 0.3638} - {0.2714{\mathbb{i}}}} & {0.3209 + {0.7763{\mathbb{i}}}} & {{- 0.1557} - {0.0019{\mathbb{i}}}} \end{matrix}$

${{V4}\left( {\text{:},\text{:},37} \right)} = \begin{matrix} {0.3841 + {0.3851{\mathbb{i}}}} & {0.1371 + {0.7838{\mathbb{i}}}} & {0.2479 + {0.0123{\mathbb{i}}}} & {{- 0.0008} - {0.0975{\mathbb{i}}}} \\ {{- 0.2285} - {0.6580{\mathbb{i}}}} & {{- 0.0956} + {0.3657{\mathbb{i}}}} & {0.2440 + {0.3447{\mathbb{i}}}} & {{- 0.4391} - {0.0271{\mathbb{i}}}} \\ {0.3448 + {0.0735{\mathbb{i}}}} & {{- 0.1099} - {0.0181{\mathbb{i}}}} & {{- 0.2141} + {0.1937{\mathbb{i}}}} & {{- 0.2988} + {0.8311{\mathbb{i}}}} \\ {{- 0.0056} + {0.3076{\mathbb{i}}}} & {0.3423 - {0.3074{\mathbb{i}}}} & {0.2247 + {0.7913{\mathbb{i}}}} & {{- 0.0349} - {0.1254{\mathbb{i}}}} \end{matrix}$ ${{V4}\left( {\text{:},\text{:},38} \right)} = \begin{matrix} {0.4422 + {0.0928{\mathbb{i}}}} & {{- 0.0860} - {0.2241{\mathbb{i}}}} & {0.5338 - {0.0441{\mathbb{i}}}} & {{- 0.1033} + {0.6639{\mathbb{i}}}} \\ {{- 0.0137} - {0.2102{\mathbb{i}}}} & {{- 0.6015} + {0.2527{\mathbb{i}}}} & {{- 0.4372} + {0.3957{\mathbb{i}}}} & {{- 0.1471} + {0.4008{\mathbb{i}}}} \\ {0.2479 + {0.5606{\mathbb{i}}}} & {{- 0.4157} - {0.2206{\mathbb{i}}}} & {0.0012 + {0.2819{\mathbb{i}}}} & {0.4968 - {0.2768{\mathbb{i}}}} \\ {0.2479 + {0.5606{\mathbb{i}}}} & {0.1543 + {0.5211{\mathbb{i}}}} & {{- 0.3515} - {0.4030{\mathbb{i}}}} & {{- 0.0551} + {0.1997{\mathbb{i}}}} \end{matrix}$ ${{V4}\left( {\text{:},\text{:},39} \right)} = \begin{matrix} {0.0337 + {0.6193{\mathbb{i}}}} & {{- 0.1081} + {0.4460{\mathbb{i}}}} & {0.0435 - {0.3239{\mathbb{i}}}} & {{- 0.1491} + {0.5251{\mathbb{i}}}} \\ {{- 0.2280} - {0.1515{\mathbb{i}}}} & {{- 0.3567} - {0.5167{\mathbb{i}}}} & {{- 0.1969} - {0.3585{\mathbb{i}}}} & {0.3858 + {0.4634{\mathbb{i}}}} \\ {0.2280 + {0.1515{\mathbb{i}}}} & {{- 0.2354} - {0.1914{\mathbb{i}}}} & {0.2532 + {0.7745{\mathbb{i}}}} & {0.0863 + {0.4020{\mathbb{i}}}} \\ {0.4621 + {0.5019{\mathbb{i}}}} & {{- 0.1985} - {0.5136{\mathbb{i}}}} & {0.0895 - {0.2322{\mathbb{i}}}} & {{- 0.0884} - {0.4019{\mathbb{i}}}} \end{matrix}$ ${{V4}\left( {\text{:},\text{:},40} \right)} = \begin{matrix} {0.0918 + {0.3270{\mathbb{i}}}} & {{- 0.5060} + {0.1443{\mathbb{i}}}} & {0.5534 - {0.3836{\mathbb{i}}}} & {{- 0.1155} - {0.3757{\mathbb{i}}}} \\ {0.4045 - {0.4815{\mathbb{i}}}} & {{- 0.4682} - {0.5571{\mathbb{i}}}} & {{- 0.1369} - {0.0572{\mathbb{i}}}} & {{- 0.2290} + {0.0234{\mathbb{i}}}} \\ {{- 0.1311} - {0.6387{\mathbb{i}}}} & {0.4345 + {0.0089{\mathbb{i}}}} & {0.5727 - {0.1978{\mathbb{i}}}} & {{- 0.1012} - {0.0932{\mathbb{i}}}} \\ {0.0541 - {0.2473{\mathbb{i}}}} & {{- 0.0309} + {0.0618{\mathbb{i}}}} & {{- 0.2903} - {0.2707{\mathbb{i}}}} & {0.7177 - {0.5085{\mathbb{i}}}} \end{matrix}$

${{V4}\left( {\text{:},\text{:},41} \right)} = \begin{matrix} {0.3841 + {0.3851{\mathbb{i}}}} & {0.0427 + {0.4037{\mathbb{i}}}} & {0.1038 + {0.3577{\mathbb{i}}}} & {0.5172 + {0.3648{\mathbb{i}}}} \\ {0.6580 - {0.2285{\mathbb{i}}}} & {{- 0.3221} + {0.3130{\mathbb{i}}}} & {{- 0.3384} - {0.3674{\mathbb{i}}}} & {{- 0.2477} + {0.0474{\mathbb{i}}}} \\ {{- 0.3448} - {0.0735{\mathbb{i}}}} & {{- 0.0724} + {0.1313{\mathbb{i}}}} & {{- 0.0231} - {0.7092{\mathbb{i}}}} & {0.5412 + {0.2385{\mathbb{i}}}} \\ {0.3076 + {0.0056{\mathbb{i}}}} & {0.3259 - {0.7104{\mathbb{i}}}} & {{- 0.3205} - {0.0747{\mathbb{i}}}} & {0.1184 + {0.4148{\mathbb{i}}}} \end{matrix}$ ${{V4}\left( {\text{:},\text{:},42} \right)} = \begin{matrix} {0.4422 + {0.1928{\mathbb{i}}}} & {{- 0.1512} + {0.0220{\mathbb{i}}}} & {{- 0.5578} + {0.0360{\mathbb{i}}}} & {{- 0.4803} + {0.4790{\mathbb{i}}}} \\ {0.2102 - {0.0137{\mathbb{i}}}} & {{- 0.5297} - {0.2125{\mathbb{i}}}} & {{- 0.4245} + {0.2405{\mathbb{i}}}} & {0.3258 - {0.5345{\mathbb{i}}}} \\ {{- 0.2479} - {0.5606{\mathbb{i}}}} & {0.3488 + {0.2574{\mathbb{i}}}} & {{- 0.5660} - {0.2718{\mathbb{i}}}} & {{- 0.0021} - {0.2051{\mathbb{i}}}} \\ {0.5606 - {0.2479{\mathbb{i}}}} & {0.5345 - {0.4211{\mathbb{i}}}} & {0.1552 + {0.1767{\mathbb{i}}}} & {{- 0.1657} - {0.2802{\mathbb{i}}}} \end{matrix}$ ${{V4}\left( {\text{:},\text{:},43} \right)} = \begin{matrix} {0.3841 + {0.3851{\mathbb{i}}}} & {0.3899 + {0.5099{\mathbb{i}}}} & {0.1239 + {0.3480{\mathbb{i}}}} & {{- 0.3212} + {0.2294{\mathbb{i}}}} \\ {0.0022 + {0.1690{\mathbb{i}}}} & {{- 0.2642} - {0.2730{\mathbb{i}}}} & {{- 0.0709} + {0.7215{\mathbb{i}}}} & {0.4401 + {0.3283{\mathbb{i}}}} \\ {{- 0.3076} - {0.0056{\mathbb{i}}}} & {0.0634 + {0.5454{\mathbb{i}}}} & {{- 0.0400} - {0.3119{\mathbb{i}}}} & {0.5426 + {0.4590{\mathbb{i}}}} \\ {{- 0.7536} - {0.1140{\mathbb{i}}}} & {0.3743 - {0.0464{\mathbb{i}}}} & {0.3735 + {0.3156{\mathbb{i}}}} & {{- 0.1921} - {0.0291{\mathbb{i}}}} \end{matrix}$ ${{V4}\left( {\text{:},\text{:},44} \right)} = \begin{matrix} {{- 0.1560} + {0.4926{\mathbb{i}}}} & {0.1996 + {0.3349{\mathbb{i}}}} & {0.3567 - {0.6608{\mathbb{i}}}} & {0.1085 + {0.0728{\mathbb{i}}}} \\ {0.0837 + {0.4175{\mathbb{i}}}} & {0.1647 + {0.4478{\mathbb{i}}}} & {{- 0.0991} + {0.3844{\mathbb{i}}}} & {{- 0.5490} - {0.3633{\mathbb{i}}}} \\ {{- 0.4597} + {0.3989{\mathbb{i}}}} & {0.0029 - {0.0825{\mathbb{i}}}} & {{- 0.4340} + {0.1941{\mathbb{i}}}} & {{- 0.0351} + {0.6288{\mathbb{i}}}} \\ {{- 0.4175} + {0.0837{\mathbb{i}}}} & {0.6245 - {0.4727{\mathbb{i}}}} & {0.1497 + {0.1732{\mathbb{i}}}} & {0.1381 - {0.3657{\mathbb{i}}}} \end{matrix}$

${{V4}\left( {\text{:},\text{:},45} \right)} = \begin{matrix} {0.3841 + {0.3851{\mathbb{i}}}} & {0.4945 - {0.2660{\mathbb{i}}}} & {{- 0.4949} + {0.3789{\mathbb{i}}}} & {{- 0.0039} - {0.0184{\mathbb{i}}}} \\ {{- 0.1140} + {0.7536{\mathbb{i}}}} & {{- 0.4531} - {0.0810{\mathbb{i}}}} & {0.2332 + {0.1714{\mathbb{i}}}} & {{- 0.0493} - {0.3478{\mathbb{i}}}} \\ {{- 0.3076} - {0.0056{\mathbb{i}}}} & {0.2295 + {0.4493{\mathbb{i}}}} & {0.2178 + {0.6234{\mathbb{i}}}} & {{- 0.4191} + {0.1977{\mathbb{i}}}} \\ {{- 0.1690} + {0.0022{\mathbb{i}}}} & {0.0234 - {0.4666{\mathbb{i}}}} & {{- 0.0755} - {0.2931{\mathbb{i}}}} & {{- 0.8112} + {0.0581{\mathbb{i}}}} \end{matrix}$ ${{V4}\left( {\text{:},\text{:},46} \right)} = \begin{matrix} {0.3396 + {0.1614{\mathbb{i}}}} & {{- 0.0891} - {0.1737`{\mathbb{i}}}} & {{- 0.5874} + {0.0189{\mathbb{i}}}} & {0.4181 + {0.5480{\mathbb{i}}}} \\ {{- 0.3400} - {0.3060{\mathbb{i}}}} & {{- 0.2160} + {0.0403{\mathbb{i}}}} & {{- 0.4608} - {0.5955{\mathbb{i}}}} & {0.2155 - {0.3594{\mathbb{i}}}} \\ {0.3493 - {0.5641{\mathbb{i}}}} & {0.3530 + {0.5446{\mathbb{i}}}} & {{- 0.2058} - {0.0338{\mathbb{i}}}} & {{- 0.2629} + {0.1610{\mathbb{i}}}} \\ {{- 0.3060} + {0.3400{\mathbb{i}}}} & {0.1657 + {0.6818{\mathbb{i}}}} & {0.1876 - {0.0948{\mathbb{i}}}} & {0.4723 + {0.1766{\mathbb{i}}}} \end{matrix}$ ${{V4}\left( {\text{:},\text{:},47} \right)} = \begin{matrix} {0.3841 + {0.3851{\mathbb{i}}}} & {0.3162 - {0.1330{\mathbb{i}}}} & {{- 0.5287} - {0.1679{\mathbb{i}}}} & {0.1564 + {0.5043{\mathbb{i}}}} \\ {{- 0.1690} + {0.0022{\mathbb{i}}}} & {{- 0.2451} + {0.4651{\mathbb{i}}}} & {0.2831 + {0.1482{\mathbb{i}}}} & {0.4827 + {0.6000{\mathbb{i}}}} \\ {0.3076 + {0.0056{\mathbb{i}}}} & {0.1535 - {0.5047{\mathbb{i}}}} & {0.7325 - {0.1862{\mathbb{i}}}} & {0.0058 + {0.2362{\mathbb{i}}}} \\ {{- 0.1140} + {0.7536{\mathbb{i}}}} & {{- 0.4972} - {0.2836{\mathbb{i}}}} & {0.0497 + {0.1282{\mathbb{i}}}} & {0.1692 - {0.2094{\mathbb{i}}}} \end{matrix}$ ${{V4}\left( {\text{:},\text{:},48} \right)} = \begin{matrix} {{- 0.1560} + {0.4926{\mathbb{i}}}} & {0.1157 + {0.3948{\mathbb{i}}}} & {0.5787 - {0.3292{\mathbb{i}}}} & {0.2943 - {0.1843{\mathbb{i}}}} \\ {{- 0.4175} + {0.0837{\mathbb{i}}}} & {0.2814 - {0.3175{\mathbb{i}}}} & {0.2525 - {0.2620{\mathbb{i}}}} & {{- 0.5641} + {0.4338{\mathbb{i}}}} \\ {0.4597 - {0.3989{\mathbb{i}}}} & {0.4393 + {0.0781{\mathbb{i}}}} & {0.3512 + {0.0363{\mathbb{i}}}} & {0.2819 + {0.4758{\mathbb{i}}}} \\ {0.0837 + {0.4175{\mathbb{i}}}} & {0.3704 - {0.5609{\mathbb{i}}}} & {0.1336 + {0.5309{\mathbb{i}}}} & {0.1521 - {0.2100{\mathbb{i}}}} \end{matrix}$

${{V4}\left( {\text{:},\text{:},49} \right)} = \begin{matrix} {0.3841 + {0.3851{\mathbb{i}}}} & {{- 0.6183} - {0.3893{\mathbb{i}}}} & {{- 0.3104} - {0.0769{\mathbb{i}}}} & {0.2262 - {0.1301{\mathbb{i}}}} \\ {0.1140 - {0.7536{\mathbb{i}}}} & {0.0439 - {0.4301{\mathbb{i}}}} & {0.0424 - {0.0174{\mathbb{i}}}} & {0.4717 - {0.0872`{\mathbb{i}}}} \\ {{- 0.3076} - {0.0056{\mathbb{i}}}} & {{- 0.3231} + {0.0445{\mathbb{i}}}} & {{- 0.0056} + {0.4980{\mathbb{i}}}} & {0.2818 + {0.6867{\mathbb{i}}}} \\ {0.1690 - {0.0022{\mathbb{i}}}} & {{- 0.2461} - {0.3351{\mathbb{i}}}} & {0.7828 + {0.1865{\mathbb{i}}}} & {{- 0.3883} + {0.0121{\mathbb{i}}}} \end{matrix}$ ${{V4}\left( {\text{:},\text{:},50} \right)} = \begin{matrix} {0.3396 + {0.1614{\mathbb{i}}}} & {0.3566 - {0.2418{\mathbb{i}}}} & {0.0169 - {0.4765{\mathbb{i}}}} & {{- 0.5447} - {0.3859{\mathbb{i}}}} \\ {0.3060 - {0.3400{\mathbb{i}}}} & {{- 0.1924} + {0.0246{\mathbb{i}}}} & {{- 0.3480} + {0.4512{\mathbb{i}}}} & {0.0113 - {0.6545{\mathbb{i}}}} \\ {{- 0.3493} + {0.5641{\mathbb{i}}}} & {{- 0.5527} + {0.1626{\mathbb{i}}}} & {{- 0.3128} - {0.0721{\mathbb{i}}}} & {{- 0.3061} - {0.1765{\mathbb{i}}}} \\ {0.3400 + {0.3060{\mathbb{i}}}} & {{- 0.4030} - {0.5315{\mathbb{i}}}} & {0.5097 + {0.2917{\mathbb{i}}}} & {0.0150 + {0.0288{\mathbb{i}}}} \end{matrix}$ ${{V4}\left( {\text{:},\text{:},51} \right)} = \begin{matrix} {0.3841 + {0.3851{\mathbb{i}}}} & {0.0072 + {0.6080{\mathbb{i}}}} & {0.0164 + {0.4489{\mathbb{i}}}} & {0.2019 + {0.3032{\mathbb{i}}}} \\ {{- 0.0022} - {0.1690{\mathbb{i}}}} & {{- 0.1531} - {0.3405{\mathbb{i}}}} & {{- 0.1388} + {0.8192{\mathbb{i}}}} & {0.0946 - {0.3643{\mathbb{i}}}} \\ {{- 0.3076} - {0.0056{\mathbb{i}}}} & {0.4820 + {0.4643{\mathbb{i}}}} & {{- 0.2314} + {0.1231{\mathbb{i}}}} & {{- 0.4842} + {0.3928{\mathbb{i}}}} \\ {0.7536 + {0.1140{\mathbb{i}}}} & {{- 0.1738} - {0.1132{\mathbb{i}}}} & {{- 0.1247} - {0.1538{\mathbb{i}}}} & {{- 0.4937} - {0.3051{\mathbb{i}}}} \end{matrix}$ ${{V4}\left( {\text{:},\text{:},52} \right)} = \begin{matrix} {{- 0.1560} + {0.4926{\mathbb{i}}}} & {{- 0.4387} - {0.0343{\mathbb{i}}}} & {0.5100 - {0.0764{\mathbb{i}}}} & {0.2952 + {0.4317{\mathbb{i}}}} \\ {0.4175 - {0.0837`{\mathbb{i}}}} & {0.5062 - {0.5972{\mathbb{i}}}} & {0.2808 - {0.2160{\mathbb{i}}}} & {0.1570 + {0.2361{\mathbb{i}}}} \\ {0.4597 - {0.3989{\mathbb{i}}}} & {{- 0.2611} + {0.2853{\mathbb{i}}}} & {{- 0.1195} + {0.0730{\mathbb{i}}}} & {{- 0.1784} + {0.6546{\mathbb{i}}}} \\ {{- 0.0837} - {0.4175{\mathbb{i}}}} & {{- 0.0591} + {0.2011{\mathbb{i}}}} & {0.7066 - {0.2995{\mathbb{i}}}} & {{- 0.3568} - {0.2417{\mathbb{i}}}} \end{matrix}$

${{V4}\left( {\text{:},\text{:},53} \right)} = \begin{matrix} {0.3841 + {0.3851{\mathbb{i}}}} & {{- 0.0757} - {0.5813{\mathbb{i}}}} & {0.1457 - {0.0933{\mathbb{i}}}} & {{- 0.5106} + {0.2643{\mathbb{i}}}} \\ {0.7536 + {0.1140{\mathbb{i}}}} & {0.1019 + {0.2454{\mathbb{i}}}} & {0.3241 + {0.2186{\mathbb{i}}}} & {0.2092 - {0.3897{\mathbb{i}}}} \\ {0.3076 + {0.0056{\mathbb{i}}}} & {0.5989 + {0.2606{\mathbb{i}}}} & {{- 0.5728} - {0.0137{\mathbb{i}}}} & {{- 0.1198} + {0.3689{\mathbb{i}}}} \\ {{- 0.0022} - {0.1690{\mathbb{i}}}} & {{- 0.3284} + {0.2264{\mathbb{i}}}} & {{- 0.1976} + {0.6708{\mathbb{i}}}} & {{- 0.5620} - {0.0868{\mathbb{i}}}} \end{matrix}$ ${{V4}\left( {\text{:},\text{:},54} \right)} = \begin{matrix} {0.3396 + {0.1614{\mathbb{i}}}} & {0.2711 - {0.7070{\mathbb{i}}}} & {{- 0.2911} + {0.2527{\mathbb{i}}}} & {{- 0.1768} + {0.3248{\mathbb{i}}}} \\ {0.3400 + {0.3060{\mathbb{i}}}} & {{- 0.4123} - {0.1234{\mathbb{i}}}} & {0.0110 - {0.0641{\mathbb{i}}}} & {{- 0.4612} - {0.6234{\mathbb{i}}}} \\ {0.3493 - {0.5641{\mathbb{i}}}} & {0.4714 - {0.0224{\mathbb{i}}}} & {0.3098 + {0.2417{\mathbb{i}}}} & {0.0189 - {0.4271{\mathbb{i}}}} \\ {0.3060 - {0.3400{\mathbb{i}}}} & {{- 0.0617} - {0.1224{\mathbb{i}}}} & {{- 0.4468} - {0.7023{\mathbb{i}}}} & {0.2720 - {0.0718{\mathbb{i}}}} \end{matrix}$

5. Fifth Scheme to Design a 6-Bit Codebook

According to an exemplary embodiment, a 4-bit codebook and a 6-bit codebook may be provided in an integrated form.

(1) Operation 1:

16 matrices may be given as follows:

${W\left( {\text{:},\text{:},1} \right)} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 & 0.5000 \\ 0.5000 & {0.0000 + {0.5000{\mathbb{i}}}} & {{- 0.5000} + {0.0000{\mathbb{i}}}} & {{- 0.0000} - {0.5000{\mathbb{i}}}} \\ 0.5000 & {{- 0.5000} + {0.0000{\mathbb{i}}}} & {0.5000 - {0.0000{\mathbb{i}}}} & {{- 0.5000} + {0.0000{\mathbb{i}}}} \\ 0.5000 & {{- 0.0000} - {0.5000{\mathbb{i}}}} & {{- 0.5000} + {0.0000{\mathbb{i}}}} & {0.0000 + {0.5000{\mathbb{i}}}} \end{matrix}$ ${W\left( {\text{:},\text{:},2} \right)} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 & 0.5000 \\ {0.3536 + {0.3536{\mathbb{i}}}} & {{- 0.3536} + {0.3536{\mathbb{i}}}} & {{- 0.3536} - {0.3536{\mathbb{i}}}} & {0.3536 - {0.3536{\mathbb{i}}}} \\ {0.0000 + {0.5000{\mathbb{i}}}} & {{- 0.0000} - {0.5000{\mathbb{i}}}} & {0.0000 + {0.5000{\mathbb{i}}}} & {{- 0.0000} - {0.5000{\mathbb{i}}}} \\ {{- 0.3536} + {0.3536{\mathbb{i}}}} & {0.3536 + {0.3536{\mathbb{i}}}} & {0.3536 - {0.3536{\mathbb{i}}}} & {{- 0.3536} - {0.3536{\mathbb{i}}}} \end{matrix}$ ${W\left( {\text{:},\text{:},3} \right)} = \begin{matrix} {0.4619 + {0.1913{\mathbb{i}}}} & {{- 0.1913} + {0.4619{\mathbb{i}}}} & {{- 0.4619} - {0.1913{\mathbb{i}}}} & {0.1913 - {0.4619{\mathbb{i}}}} \\ {0.4455 - {0.2270{\mathbb{i}}}} & {0.4455 - {0.2270{\mathbb{i}}}} & {0.4455 - {0.2270{\mathbb{i}}}} & {0.4455 - {0.2270{\mathbb{i}}}} \\ {{- 0.1167} + {0.4862{\mathbb{i}}}} & {0.4862 + {0.1167{\mathbb{i}}}} & {0.1167 - {0.4862{\mathbb{i}}}} & {{- 0.4862} - {0.1167{\mathbb{i}}}} \\ {0.2939 + {0.4045{\mathbb{i}}}} & {{- 0.2939} - {0.4045{\mathbb{i}}}} & {0.2939 + {0.4045{\mathbb{i}}}} & {{- 0.2939} - {0.4045{\mathbb{i}}}} \end{matrix}$ ${W\left( {\text{:},\text{:},4} \right)} = \begin{matrix} {- 0.5000} & {- 0.5000} & {- 0.5000`} & {- 0.5000} \\ {{- 0.4985} - {0.0392{\mathbb{i}}}} & {0.0392 + {0.4985{\mathbb{i}}}} & {{- 0.4985} + {0.0392{\mathbb{i}}}} & {{- 0.0392} - {0.4985{\mathbb{i}}}} \\ {0.4938 - {0.0782{\mathbb{i}}}} & {{- 0.4938} + {0.0782{\mathbb{i}}}} & {0.4938 - {0.0782{\mathbb{i}}}} & {{- 0.4938} + {0.0782{\mathbb{i}}}} \\ {{- 0.4862} + {0.1167{\mathbb{i}}}} & {0.1167 + {0.4862{\mathbb{i}}}} & {0.4862 - {0.1167{\mathbb{i}}}} & {{- 0.1167} - {0.4862{\mathbb{i}}}} \end{matrix}$

${W\left( {\text{:},\text{:},5} \right)} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 & 0.5000 \\ {0.4619 + {0.1913{\mathbb{i}}}} & {{- 0.1913} + {0.4619{\mathbb{i}}}} & {{- 0.4619} - {0.1913{\mathbb{i}}}} & {0.1913 - {0.4619{\mathbb{i}}}} \\ {0.3536 + {0.3536{\mathbb{i}}}} & {{- 0.3536} - {0.3536{\mathbb{i}}}} & {0.3536 + {0.3536{\mathbb{i}}}} & {{- 0.3536} - {0.3536{\mathbb{i}}}} \\ {0.1913 + {0.4619{\mathbb{i}}}} & {0.4619 - {0.1913{\mathbb{i}}}} & {{- 0.1913} - {0.4619{\mathbb{i}}}} & {{- 0.4619} + {0.1913{\mathbb{i}}}} \end{matrix}$ ${W\left( {\text{:},\text{:},6} \right)} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 & 0.5000 \\ {0.1913 + {0.4619{\mathbb{i}}}} & {{- 0.4619} + {0.1913{\mathbb{i}}}} & {{- 0.1913} - {0.4619{\mathbb{i}}}} & {0.4619 - {0.1913{\mathbb{i}}}} \\ {{- 0.3536} + {0.3536{\mathbb{i}}}} & {0.3536 - {0.3536{\mathbb{i}}}} & {{- 0.3536} + {0.3536{\mathbb{i}}}} & {0.3536 - {0.3536{\mathbb{i}}}} \\ {{- 0.4619} - {0.1913{\mathbb{i}}}} & {{- 0.1913} + {0.4619{\mathbb{i}}}} & {0.4619 + {0.1913{\mathbb{i}}}} & {0.1913 - {0.4619{\mathbb{i}}}} \end{matrix}$ ${W\left( {\text{:},\text{:},7} \right)} = \begin{matrix} {0.2437 + {0.4837{\mathbb{i}}}} & {0.4258 + {0.0076{\mathbb{i}}}} & {0.5210 - {0.4265{\mathbb{i}}}} & {{- 0.2364} - {0.1268{\mathbb{i}}}} \\ {0.5740 - {0.3102{\mathbb{i}}}} & {0.0596 + {0.3222{\mathbb{i}}}} & {0.2922 + {0.0454{\mathbb{i}}}} & {0.6147 + {0.0409{\mathbb{i}}}} \\ {0.2823 - {0.3783{\mathbb{i}}}} & {{- 0.4186} + {0.1136{\mathbb{i}}}} & {0.3628 + {0.1522{\mathbb{i}}}} & {{- 0.6500} - {0.1085{\mathbb{i}}}} \\ {0.2062 + {0.1248{\mathbb{i}}}} & {0.1513 + {0.7073{\mathbb{i}}}} & {{- 0.5498} - {0.0464{\mathbb{i}}}} & {{- 0.2340} - {0.2440{\mathbb{i}}}} \end{matrix}$ ${W\left( {\text{:},\text{:},8} \right)} = \begin{matrix} {0.0591 - {0.4953{\mathbb{i}}}} & {0.0742 + {0.2714{\mathbb{i}}}} & {0.3370 + {0.2000{\mathbb{i}}}} & {{- 0.1041} - {0.7125{\mathbb{i}}}} \\ {0.5545 + {0.3908{\mathbb{i}}}} & {{- 0.0591} - {0.0963{\mathbb{i}}}} & {0.2158 + {0.4089{\mathbb{i}}}} & {{- 0.5589} + {0.0299{\mathbb{i}}}} \\ {0.2593 - {0.0234{\mathbb{i}}}} & {{- 0.5800} - {0.1645{\mathbb{i}}}} & {{- 0.2363} - {0.5965{\mathbb{i}}}} & {{- 0.2149} - {0.3331{\mathbb{i}}}} \\ {0.3300 + {0.3380{\mathbb{i}}}} & {0.4619 + {0.5755{\mathbb{i}}}} & {0.0175 - {0.4698{\mathbb{i}}}} & {0.0748 - {0.0748{\mathbb{i}}}} \end{matrix}$

${W\left( {\text{:},\text{:},9} \right)} = \begin{matrix} {0.5019 - {0.2171{\mathbb{i}}}} & {0.2121 - {0.4391{\mathbb{i}}}} & {0.4649 + {0.1545{\mathbb{i}}}} & {{- 0.4709} + {0.0383{\mathbb{i}}}} \\ {0.0570 + {0.3132{\mathbb{i}}}} & {0.3637 + {0.1907{\mathbb{i}}}} & {0.5971 - {0.5122{\mathbb{i}}}} & {0.3285 + {0.0569{\mathbb{i}}}} \\ {0.2569 - {0.1797{\mathbb{i}}}} & {0.0539 - {0.3157{\mathbb{i}}}} & {{- 0.0025} + {0.3671{\mathbb{i}}}} & {0.8042 + {0.1327{\mathbb{i}}}} \\ {0.6949 + {0.1358{\mathbb{i}}}} & {{- 0.5835} + {0.3879{\mathbb{i}}}} & {0.0058 - {0.0797{\mathbb{i}}}} & {0.0341 + {0.0124{\mathbb{i}}}} \end{matrix}$ ${W\left( {\text{:},\text{:},10} \right)} = \begin{matrix} {0.2199 + {0.3199{\mathbb{i}}}} & {0.6305 - {0.4345{\mathbb{i}}}} & {{- 0.0525} + {0.2357{\mathbb{i}}}} & {{- 0.3110} - {0.3286{\mathbb{i}}}} \\ {0.5983 + {0.1264{\mathbb{i}}}} & {{- 0.0683} + {0.2985{\mathbb{i}}}} & {0.5833 + {0.2614{\mathbb{i}}}} & {0.2969 - {0.1888{\mathbb{i}}}} \\ {{- 0.5576} - {0.2903{\mathbb{i}}}} & {{- 0.0495} - {0.3235{\mathbb{i}}}} & {0.3552 + {0.3750{\mathbb{i}}}} & {0.2888 - {0.3841{\mathbb{i}}}} \\ {{- 0.0940} - {0.2672{\mathbb{i}}}} & {{- 0.0481} + {0.4588{\mathbb{i}}}} & {0.0118 + {0.5160{\mathbb{i}}}} & {{- 0.6633} - {0.0260{\mathbb{i}}}} \end{matrix}$ ${W\left( {\text{:},\text{:},11} \right)} = \begin{matrix} {0.4534 + {0.1075{\mathbb{i}}}} & {0.2632 + {0.4348{\mathbb{i}}}} & {0.1838 + {0.6429{\mathbb{i}}}} & {0.2307 + {0.1558{\mathbb{i}}}} \\ {{- 0.0163} + {0.7556{\mathbb{i}}}} & {0.3698 - {0.0222{\mathbb{i}}}} & {0.0055 - {0.3112{\mathbb{i}}}} & {0.3467 - {0.2728{\mathbb{i}}}} \\ {0.0681 + {0.3202{\mathbb{i}}}} & {{- 0.4066} - {0.6110{\mathbb{i}}}} & {0.3882 + {0.2198{\mathbb{i}}}} & {0.1705 + {0.3551{\mathbb{i}}}} \\ {0.1259 + {0.2977{\mathbb{i}}}} & {0.0347 + {0.2542{\mathbb{i}}}} & {0.4482 - {0.2369{\mathbb{i}}}} & {{- 0.7319} + {0.1924{\mathbb{i}}}} \end{matrix}$ ${W\left( {\text{:},\text{:},12} \right)} = \begin{matrix} {0.0534 - {0.5633{\mathbb{i}}}} & {{- 0.4493} + {0.3172{\mathbb{i}}}} & {{- 0.3346} + {0.1479{\mathbb{i}}}} & {{- 0.4868} - {0.0811{\mathbb{i}}}} \\ {{- 0.1472} - {0.1184{\mathbb{i}}}} & {{- 0.5422} + {0.3203{\mathbb{i}}}} & {0.2302 - {0.4770{\mathbb{i}}}} & {0.4913 + {0.2140{\mathbb{i}}}} \\ {{- 0.7445} + {0.0533{\mathbb{i}}}} & {{- 0.1168} - {0.4073{\mathbb{i}}}} & {{- 0.1288} - {0.3642{\mathbb{i}}}} & {{- 0.2957} - {0.1634{\mathbb{i}}}} \\ {0.2881 - {0.0638{\mathbb{i}}}} & {0.3438 + {0.0562{\mathbb{i}}}} & {{- 0.1358} - {0.6465{\mathbb{i}}}} & {{- 0.3579} + {0.4766{\mathbb{i}}}} \end{matrix}$

${W\left( {\text{:},\text{:},13} \right)} = \begin{matrix} {0.4192 + {0.3317{\mathbb{i}}}} & {0.6462 - {0.3049{\mathbb{i}}}} & {0.1765 - {0.0924{\mathbb{i}}}} & {0.3680 + {0.1691{\mathbb{i}}}} \\ {{- 0.2340} + {0.0529{\mathbb{i}}}} & {{- 0.0388} - {0.4789{\mathbb{i}}}} & {0.6212 - {0.0425{\mathbb{i}}}} & {{- 0.5223} + {0.2262{\mathbb{i}}}} \\ {0.0491 - {0.2199{\mathbb{i}}}} & {{- 0.2962} + {0.1142{\mathbb{i}}}} & {0.5130 + {0.5000{\mathbb{i}}}} & {0.5366 + {0.2175{\mathbb{i}}}} \\ {{- 0.7746} + {0.0772{\mathbb{i}}}} & {0.1089 - {0.3821{\mathbb{i}}}} & {{- 0.2395} + {0.0459{\mathbb{i}}}} & {0.4196 + {0.0255{\mathbb{i}}}} \end{matrix}$ ${W\left( {\text{:},\text{:},14} \right)} = \begin{matrix} {0.0848 + {0.2248`{\mathbb{i}}}} & {0.5883 + {0.1906{\mathbb{i}}}} & {{- 0.4550} - {0.1147{\mathbb{i}}}} & {{- 0.5202} + {0.2628{\mathbb{i}}}} \\ {{- 0.2010} + {0.7357{\mathbb{i}}}} & {{- 0.0481} - {0.2343{\mathbb{i}}}} & {{- 0.3923} + {0.0511{\mathbb{i}}}} & {0.3107 - {0.3288{\mathbb{i}}}} \\ {0.3862 + {0.2270{\mathbb{i}}}} & {0.2323 + {0.5375{\mathbb{i}}}} & {0.1654 - {0.1078{\mathbb{i}}}} & {0.6089 + {0.2161{\mathbb{i}}}} \\ {{- 0.3114} + {0.2509{\mathbb{i}}}} & {0.4661 + {0.0138{\mathbb{i}}}} & {0.6358 + {0.4245{\mathbb{i}}}} & {{- 0.1325} - {0.1438{\mathbb{i}}}} \end{matrix}$ ${W\left( {\text{:},\text{:},15} \right)} = \begin{matrix} {0.3968 - {0.0671{\mathbb{i}}}} & {0.3110 + {0.1356{\mathbb{i}}}} & {{- 0.2168} - {0.4540{\mathbb{i}}}} & {{- 0.6008} - {0.3299{\mathbb{i}}}} \\ {{- 0.8289} - {0.0667{\mathbb{i}}}} & {{- 0.2015} + {0.0416{\mathbb{i}}}} & {0.1102 - {0.2871{\mathbb{i}}}} & {{- 0.4011} - {0.1031{\mathbb{i}}}} \\ {0.1770 - {0.1687{\mathbb{i}}}} & {{- 0.3325} - {0.1370{\mathbb{i}}}} & {0.0567 - {0.7634{\mathbb{i}}}} & {0.2785 + {0.3837{\mathbb{i}}}} \\ {{- 0.2814} + {0.0866{\mathbb{i}}}} & {0.7347 - {0.4165{\mathbb{i}}}} & {{- 0.0390} - {0.2544{\mathbb{i}}}} & {0.3519 - {0.1000{\mathbb{i}}}} \end{matrix}$ ${W\left( {\text{:},\text{:},16} \right)} = \begin{matrix} {0.0743 - {0.7846{\mathbb{i}}}} & {{- 0.4063} + {0.3394{\mathbb{i}}}} & {{- 0.2621} + {0.0798{\mathbb{i}}}} & {0.1421 + {0.0583{\mathbb{i}}}} \\ {0.1307 - {0.3436{\mathbb{i}}}} & {0.2466 + {0.0922{\mathbb{i}}}} & {0.4783 - {0.5634{\mathbb{i}}}} & {{- 0.0799} - {0.4928{\mathbb{i}}}} \\ {{- 0.3387} - {0.2385{\mathbb{i}}}} & {{- 0.4080} - {0.5055{\mathbb{i}}}} & {0.5212 + {0.1951{\mathbb{i}}}} & {{- 0.2917} + {0.1078{\mathbb{i}}}} \\ {{- 0.0962} + {0.2508{\mathbb{i}}}} & {{- 0.4767} - {0.0347{\mathbb{i}}}} & {{- 0.2418} + {0.1026{\mathbb{i}}}} & {{- 0.0231} - {0.7937{\mathbb{i}}}} \end{matrix}$

(2) Operation 2:

According to an exemplary embodiment, a 4-bit rank 1 codebook, a rank 2 codebook, a rank 3 codebook, and a rank 4 codebook, and a 6-bit rank 1 codebook, a rank 2 codebook, a rank 3 codebook, and a rank 4 codebook may be generated as given by the following Table 15 and Table 16, using column subsets of the aforementioned 16 matrices. The following Table 15 shows 6 bits of codebooks:

Codebook Transmission Transmission Transmission Transmission Index Rank 1 Rank 2 Rank 3 Rank 4 1 W(:,1,1) W(:,[1,2],1) W(:,[1,2,3],1) W(:,:,1) 2 W(:,2,1) W(:,[1,3],1) W(:,[1,2,4],1) W(:,:,2) 3 W(:,3,1) W(:,[1,4],1) W(:,[1,3,4],1) W(:,:,3) 4 W(:,4,1) W(:,[2,3],1) W(:,[2,3,4],1) W(:,:,4) 5 W(:,1,2) W(:,[2,4],1) W(:,[1,2,3],2) W(:,:,5) 6 W(:,2,2) W(:,[3,4],1) W(:,[1,2,4],2) W(:,:,6) 7 W(:,3,2) W(:,[1,2],2) W(:,[1,3,4],2) W(:,:,7) 8 W(:,4,2) W(:,[1,3],2) W(:,[2,3,4],2) W(:,:,8) 9 W(:,1,3) W(:,[1,4],2) W(:,[1,2,3],3) W(:,:,9) 10 W(:,2,3) W(:,[2,3],2) W(:,[1,2,4],3) W(:,:,10) 11 W(:,3,3) W(:,[2,4],2) W(:,[1,3,4],3) W(:,:,11) 12 W(:,4,3) W(:,[3,4],2) W(:,[2,3,4],3) W(:,:,12) 13 W(:,1,4) W(:,[1,4],3) W(:,[1,2,3],4) W(:,:,13) 14 W(:,2,4) W(:,[2,3],3) W(:,[1,2,4],4) W(:,:,14) 15 W(:,3,4) W(:,[1,2],4) W(:,[1,3,4],4) W(:,:,15) 16 W(:,4,4) W(:,[1,4],4) W(:,[2,3,4],4) W(:,:,16) 17 W(:,1,5) W(:,[1,2],5) W(:,[1,2,3],5) 18 W(:,2,5) W(:,[1,3],6) W(:,[1,2,4],5) 19 W(:,3,5) W(:,[2,3],6) W(:,[1,3,4],5) 20 W(:,4,5) W(:,[2,4],6) W(:,[2,3,4],5) 21 W(:,1,6) W(:,[1,2],7) W(:,[1,2,3],6) 22 W(:,2,6) W(:,[1,3],7) W(:,[1,2,4],6) 23 W(:,3,6) W(:,[1,4],7) W(:,[1,3,4],6) 24 W(:,4,6) W(:,[2,3],7) W(:,[2,3,4],6) 25 W(:,1,7) W(:,[2,4],7) W(:,[1,2,3],7) 26 W(:,2,7) W(:,[3,4],7) W(:,[1,2,4],7) 27 W(:,3,7) W(:,[1,2],8) W(:,[1,3,4],7) 28 W(:,4,7) W(:,[1,3],8) W(:,[2,3,4],7) 29 W(:,1,8) W(:,[1,4],8) W(:,[1,2,3],8) 30 W(:,2,8) W(:,[2,3],8) W(:,[1,2,4],8) 31 W(:,3,8) W(:,[2,4],8) W(:,[1,3,4],8) 32 W(:,4,8) W(:,[3,4],8) W(:,[2,3,4],8) 32 W(:,1,9) W(:,[1,2],9) W(:,[1,2,3],9) 34 W(:,2,9) W(:,[1,3],9) W(:,[1,2,4],9) 35 W(:,3,9) W(:,[2,3],9) W(:,[1,3,4],9) 36 W(:,4,9) W(:,[2,4],9) W(:,[2,3,4],9) 37 W(:,1,10) W(:,[1,3],10) W(:,[1,2,3],10) 38 W(:,2,10) W(:,[2,4],10) W(:,[1,2,4],10) 39 W(:,3,10) W(:,[3,4],10) W(:,[1,3,4],10) 40 W(:,4,10) W(:,[1,2],11) W(:,[2,3,4],10) 41 W(:,1,11) W(:,[1,3],11) W(:,[1,2,3],11) 42 W(:,2,11) W(:,[1,4],11) W(:,[1,2,4],11) 43 W(:,3,11) W(:,[2,3],11) W(:,[1,3,4],11) 44 W(:,4,11) W(:,[2,4],11) W(:,[2,3,4],11) 45 W(:,1,12) W(:,[3,4],11) W(:,[1,2,3],12) 46 W(:,2,12) W(:,[1,2],12) W(:,[1,2,4],12) 47 W(:,3,12) W(:,[1,3],12) W(:,[1,3,4],12) 48 W(:,4,12) W(:,[2,3],12) W(:,[2,3,4],12) 49 W(:,1,13) W(:,[1,2],13) W(:,[1,2,3],13) 50 W(:,2,13) W(:,[1,4],13) W(:,[1,2,4],13) 51 W(:,3,13) W(:,[2,4],13) W(:,[1,3,4],13) 52 W(:,4,13) W(:,[3,4],13) W(:,[2,3,4],13) 53 W(:,1,14) W(:,[1,2],14) W(:,[1,2,3],14) 54 W(:,2,14) W(:,[1,3],14) W(:,[1,2,4],14) 55 W(:,3,14) W(:,[1,4],14) W(:,[1,3,4],14) 56 W(:,4,14) W(:,[2,3],14) W(:,[2,3,4],14) 57 W(:,1,15) W(:,[2,4],14) W(:,[1,2,3],15) 58 W(:,2,15) W(:,[3,4],14) W(:,[1,2,4],15) 59 W(:,3,15) W(:,[1,4],15) W(:,[1,3,4],15) 60 W(:,4,15) W(:,[2,3],15) W(:,[2,3,4],15) 61 W(:,1,16) W(:,[1,3],16) W(:,[1,2,3],16) 62 W(:,2,16) W(:,[2,3],16) W(:,[1,2,4],16) 63 W(:,3,16) W(:,[2,4],16) W(:,[1,3,4],16) 64 W(:,4,16) W(:,[3,4],16) W(:,[2,3,4],16)

The following Table 16 shows 4 bits of codebooks:

Codebook Transmission Transmission Transmission Transmission Index Rank 1 Rank 2 Rank 3 Rank 4 1 W(:,1,1) W(:,[1,2],1) W(:,[1,2,3],1) W(:,:,1) 2 W(:,2,1) W(:,[1,3],1) W(:,[1,2,4],1) W(:,:,2) 3 W(:,3,1) W(:,[1,4],1) W(:,[1,3,4],1) W(:,:,3) 4 W(:,4,1) W(:,[2,3],1) W(:,[2,3,4],1) W(:,:,4) 5 W(:,1,2) W(:,[2,4],1) W(:,[1,2,3],2) 6 W(:,2,2) W(:,[3,4],1) W(:,[1,2,4],2) 7 W(:,3,2) W(:,[1,2],2) W(:,[1,3,4],2) 8 W(:,4,2) W(:,[1,3],2) W(:,[2,3,4],2) 9 W(:,1,3) W(:,[1,4],2) W(:,[1,2,3],3) 10 W(:,2,3) W(:,[2,3],2) W(:,[1,2,4],3) 11 W(:,3,3) W(:,[2,4],2) W(:,[1,3,4],3) 12 W(:,4,3) W(:,[3,4],2) W(:,[2,3,4],3) 13 W(:,1,4) W(:,[1,4],3) W(:,[1,2,3],4) 14 W(:,2,4) W(:,[2,3],3) W(:,[1,2,4],4) 15 W(:,3,4) W(:,[1,2],4) W(:,[1,3,4],4) 16 W(:,4,4) W(:,[1,4],4) W(:,[2,3,4],4)

Codes according to a MATLAB® program for obtaining the 6-bit codebooks disclosed in the above Table 15 may follow as:

-   -   6-Bit Rank 1 Codebook:

Codebook{1}(:,1,1:4)=W(:,[1:4],1); Codebook{1}(:,1,5:8)=W(:,[1:4],2); Codebook{1}(:,1,9:12)=W(:,[1:4],3); Codebook{1}(:,1,13:16)=W(:,[1:4],4); Codebook{1}(:,1,17:20)=W(:,[1:4],5); Codebook{1}(:,1,21:24)=W(:,[1:4],6); Codebook{1}(:,1,25:28)=W(:,[1:4],7); Codebook{1}(:,1,29:32)=W(:,[1:4],8); Codebook{1}(:,1,33:36)=W(:,[1:4],9); Codebook{1}(:,1,37:40)=W(:,[1:4],10); Codebook{1}(:,1,41:44)=W(:,[1:4],11); Codebook{1}(:,1,45:48)=W(:,[1:4],12); Codebook{1}(:,1,49:52)=W(:,[1:4],13); Codebook{1}(:,1,53:56)=W(:,[1:4],14); Codebook{1}(:,1,57:60)=W(:,[1:4],15); Codebook{1}(:,1,61:64)=W(:,[1:4],16);

-   -   6-Bit Rank 2 Codebook:

Codebook{2}(:,1:2,1)=W(:,[1,2],1); Codebook{2}(:,1:2,2)=W(:,[1,3],1); Codebook{2}(:,1:2,3)=W(:,[1,4],1); Codebook{2}(:,1:2,4)=W(:,[2,3],1); Codebook{2}(:,1:2,5)=W(:,[2,4],1); Codebook{2}(:,1:2,6)=W(:,[3,4],1); Codebook{2}(:,1:2,7)=W(:,[1,2],2); Codebook{2}(:,1:2,8)=W(:,[1,3],2); Codebook{2}(:,1:2,9)=W(:,[1,4],2); Codebook{2}(:,1:2,10)=W(:,[2,3],2); Codebook{2}(:,1:2,11)=W(:,[2,4],2); Codebook{2}(:,1:2,12)=W(:,[3,4],2); Codebook{2}(:,1:2,13)=W(:,[1,4],3); Codebook{2}(:,1:2,14)=W(:,[2,3],3); Codebook{2}(:,1:2,15)=W(:,[1,2],4); Codebook{2}(:,1:2,16)=W(:,[1,4],4); Codebook{2}(:,1:2,17)=W(:,[1,2],5); Codebook{2}(:,1:2,18)=W(:,[1,3],6); Codebook{2}(:,1:2,19)=W(:,[2,3],6); Codebook{2}(:,1:2,20)=W(:,[2,4],6); Codebook{2}(:,1:2,21)=W(:,[1,2],7); Codebook{2}(:,1:2,22)=W(:,[1,3],7); Codebook{2}(:,1:2,23)=W(:,[1,4],7); Codebook{2}(:,1:2,24)=W(:,[2,3],7); Codebook{2}(:,1:2,25)=W(:,[2,4],7); Codebook{2}(:,1:2,26)=W(:,[3,4],7); Codebook{2}(:,1:2,27)=W(:,[1,2],8); Codebook{2}(:,1:2,28)=W(:,[1,3],8); Codebook{2}(:,1:2,29)=W(:,[1,4],8); Codebook{2}(:,1:2,30)=W(:,[2,3],8); Codebook{2}(:,1:2,31)=W(:,[2,4],8); Codebook{2}(:,1:2,32)=W(:,[3,4],8); Codebook{2}(:,1:2,33)=W(:,[1,2],9); Codebook{2}(:,1:2,34)=W(:,[1,3],9); Codebook{2}(:,1:2,35)=W(:,[2,3],9); Codebook{2}(:,1:2,36)=W(:,[2,4],9); Codebook{2}(:,1:2,37)=W(:,[1,3],10); Codebook{2}(:,1:2,38)=W(:,[2,4],10); Codebook{2}(:,1:2,39)=W(:,[3,4],10); Codebook{2}(:,1:2,40)=W(:,[1,2],11); Codebook{2}(:,1:2,41)=W(:,[1,3],11); Codebook{2}(:,1:2,42)=W(:,[1,4],11); Codebook{2}(:,1:2,43)=W(:,[2,3],11); Codebook{2}(:,1:2,44)=W(:,[2,4],11); Codebook{2}(:,1:2,45)=W(:,[3,4],11); Codebook{2}(:,1:2,46)=W(:,[1,2],12); Codebook{2}(:,1:2,47)=W(:,[1,3],12); Codebook{2}(:,1:2,48)=W(:,[2,3],12); Codebook{2}(:,1:2,49)=W(:,[1,2],13); Codebook{2}(:,1:2,50)=W(:,[1,4],13); Codebook{2}(:,1:2,51)=W(:,[2,4],13); Codebook{2}(:,1:2,52)=W(:,[3,4],13); Codebook{2}(:,1:2,53)=W(:,[1,2],14); Codebook{2}(:,1:2,54)=W(:,[1,3],14); Codebook{2}(:,1:2,55)=W(:,[1,4],14); Codebook{2}(:,1:2,56)=W(:,[2,3],14); Codebook{2}(:,1:2,57)=W(:,[2,4],14); Codebook{2}(:,1:2,58)=W(:,[3,4],14); Codebook{2}(:,1:2,59)=W(:,[1,4],15); Codebook{2}(:,1:2,60)=W(:,[2,3],15); Codebook{2}(:,1:2,61)=W(:,[1,3],16); Codebook{2}(:,1:2,62)=W(:,[2,3],16); Codebook{2}(:,1:2,63)=W(:,[2,4],16); Codebook{2}(:,1:2,64)=W(:,[3,4],16);

-   -   6-Bit Rank 3 Codebook:

Codebook{3}(:,1:3,1)=W(:,[1,2,3],1); Codebook{3}(:,1:3,2)=W(:,[1,2,4],1); Codebook{3}(:,1:3,3)=W(:,[1,3,4],1); Codebook{3}(:,1:3,4)=W(:,[2,3,4],1); Codebook{3}(:,1:3,5)=W(:,[1,2,3],2); Codebook{3}(:,1:3,6)=W(:,[1,2,4],2); Codebook{3}(:,1:3,7)=W(:,[1,3,4],2); Codebook{3}(:,1:3,8)=W(:,[2,3,4],2); Codebook{3}(:,1:3,9)=W(:,[1,2,3],3); Codebook{3}(:,1:3,10)=W(:,[1,2,4],3); Codebook{3}(:,1:3,11)=W(:,[1,3,4],3); Codebook{3}(:,1:3,12)=W(:,[2,3,4],3); Codebook{3}(:,1:3,13)=W(:,[1,2,3],4); Codebook{3}(:,1:3,14)=W(:,[1,2,4],4); Codebook{3}(:,1:3,15)=W(:,[1,3,4],4); Codebook{3}(:,1:3,16)=W(:,[2,3,4],4); Codebook{3}(:,1:3,17)=W(:,[1,2,3],5); Codebook{3}(:,1:3,18)=W(:,[1,2,4],5); Codebook{3}(:,1:3,19)=W(:,[1,3,4],5); Codebook{3}(:,1:3,20)=W(:,[2,3,4],5); Codebook{3}(:,1:3,21)=W(:,[1,2,3],6); Codebook{3}(:,1:3,22)=W(:,[1,2,4],6); Codebook{3}(:,1:3,23)=W(:,[1,3,4],6); Codebook{3}(:,1:3,24)=W(:,[2,3,4],6); Codebook{3}(:,1:3,25)=W(:,[1,2,3],7); Codebook{3}(:,1:3,26)=W(:,[1,2,4],7); Codebook{3}(:,1:3,27)=W(:,[1,3,4],7); Codebook{3}(:,1:3,28)=W(:,[2,3,4],7); Codebook{3}(:,1:3,29)=W(:,[1,2,3],8); Codebook{3}(:,1:3,30)=W(:,[1,2,4],8); Codebook{3}(:,1:3,31)=W(:,[1,3,4],8); Codebook{3}(:,1:3,32)=W(:,[2,3,4],8); Codebook{3}(:,1:3,33)=W(:,[1,2,3],9); Codebook{3}(:,1:3,34)=W(:,[1,2,4],9); Codebook{3}(:,1:3,35)=W(:,[1,3,4],9); Codebook{3}(:,1:3,36)=W(:,[2,3,4],9); Codebook{3}(:,1:3,37)=W(:,[1,2,3],10); Codebook{3}(:,1:3,38)=W(:,[1,2,4],10); Codebook{3}(:,1:3,39)=W(:,[1,3,4],10); Codebook{3}(:,1:3,40)=W(:,[2,3,4],10); Codebook{3}(:,1:3,41)=W(:,[1,2,3],11); Codebook{3}(:,1:3,42)=W(:,[1,2,4],11); Codebook{3}(:,1:3,43)=W(:,[1,3,4],11); Codebook{3}(:,1:3,44)=W(:,[2,3,4],11); Codebook{3}(:,1:3,45)=W(:,[1,2,3],12); Codebook{3}(:,1:3,46)=W(:,[1,2,4],12); Codebook{3}(:,1:3,47)=W(:,[1,3,4],12); Codebook{3}(:,1:3,48)=W(:,[2,3,4],12); Codebook{3}(:,1:3,49)=W(:,[1,2,3],13); Codebook{3}(:,1:3,50)=W(:,[1,2,4],13); Codebook{3}(:,1:3,51)=W(:,[1,3,4],13); Codebook{3}(:,1:3,52)=W(:,[2,3,4],13); Codebook{3}(:,1:3,53)=W(:,[1,2,3],14); Codebook{3}(:,1:3,54)=W(:,[1,2,4],14); Codebook{3}(:,1:3,55)=W(:,[1,3,4],14); Codebook{3}(:,1:3,56)=W(:,[2,3,4],14); Codebook{3}(:,1:3,57)=W(:,[1,2,3],15); Codebook{3}(:,1:3,58)=W(:,[1,2,4],15); Codebook{3}(:,1:3,59)=W(:,[1,3,4],15); Codebook{3}(:,1:3,60)=W(:,[2,3,4],15); Codebook{3}(:,1:3,61)=W(:,[1,2,3],16); Codebook{3}(:,1:3,62)=W(:,[1,2,4],16); Codebook{3}(:,1:3,63)=W(:,[1,3,4],16); Codebook{3}(:,1:3,64)=W(:,[2,3,4],16);

-   -   6-Bit Rank 4 Codebook:

Codebook{4}(:,1:4,1)=W(:,[1,2,3,4],1); Codebook{4}(:,1:4,2)=W(:,[1,2,3,4],2); Codebook{4}(:,1:4,3)=W(:,[1,2,3,4],3); Codebook{4}(:,1:4,4)=W(:,[1,2,3,4],4); Codebook{4}(:,1:4,5)=W(:,[1,2,3,4],5); Codebook{4}(:,1:4,6)=W(:,[1,2,3,4],6); Codebook{4}(:,1:4,7)=W(:,[1,2,3,4],7); Codebook{4}(:,1:4,8)=W(:,[1,2,3,4],8); Codebook{4}(:,1:4,9)=W(:,[1,2,3,4],9); Codebook{4}(:,1:4,10)=W(:,[1,2,3,4],10); Codebook{4}(:,1:4,11)=W(:,[1,2,3,4],11); Codebook{4}(:,1:4,12)=W(:,[1,2,3,4],12); Codebook{4}(:,1:4,13)=W(:,[1,2,3,4],13); Codebook{4}(:,1:4,14)=W(:,[1,2,3,4],14); Codebook{4}(:,1:4,15)=W(:,[1,2,3,4],15); Codebook{4}(:,1:4,16)=W(:,[1,2,3,4],16);

Codes according to the MATLAB® program for obtaining the 4-bit codebooks disclosed in the above Table 16 may follow as:

-   -   4-Bit Rank 1 Codebook:

Codebook{1}(:,1,1:4)=W(:,[l:4],1); Codebook{1}(:,1,5:8)=W(:,[l:4],2); Codebook{1}(:,1,9:12)=W(:,[l:4],3); Codebook{1}(:,1,13:16)=W(:,[l:4],4);

-   -   4-Bit Rank 2 Codebook:

Codebook{2}(:,1:2,1)=W(:,[1,2],1); Codebook{2}(:,1:2,2)=W(:,[1,3],1); Codebook{2}(:,1:2,3)=W(:,[1,4],1); Codebook{2}(:,1:2,4)=W(:,[2,3],1); Codebook{2}(:,1:2,5)=W(:,[2,4],1); Codebook{2}(:,1:2,6)=W(:,[3,4],1); Codebook{2}(:,1:2,7)=W(:,[1,2],2); Codebook{2}(:,1:2,8)=W(:,[1,3],2); Codebook{2}(:,1:2,9)=W(:,[1,4],2); Codebook{2}(:,1:2,10)=W(:,[2,3],2); Codebook{2}(:,1:2,11)=W(:,[2,4],2); Codebook{2}(:,1:2,12)=W(:,[3,4],2); Codebook{2}(:,1:2,13)=W(:,[1,4],3); Codebook{2}(:,1:2,14)=W(:,[2,3],3); Codebook{2}(:,1:2,15)=W(:,[1,2],4); Codebook{2}(:,1:2,16)=W(:,[1,4],4);

-   -   4-Bit Rank 3 Codebook:

Codebook{3}(:,1:3,1)=W(:,[1,2,3],1); Codebook{3}(:,1:3,2)=W(:,[1,2,4],1); Codebook{3}(:,1:3,3)=W(:,[1,3,4],1); Codebook{3}(:,1:3,4)=W(:,[2,3,4],1); Codebook{3}(:,1:3,5)=W(:,[1,2,3],2); Codebook{3}(:,1:3,6)=W(:,[1,2,4],2); Codebook{3}(:,1:3,7)=W(:,[1,3,4],2); Codebook{3}(:,1:3,8)=W(:,[2,3,4],2); Codebook{3}(:,1:3,9)=W(:,[1,2,3],3); Codebook{3}(:,1:3,10)=W(:,[1,2,4],3); Codebook{3}(:,1:3,11)=W(:,[1,3,4],3); Codebook{3}(:,1:3,12)=W(:,[2,3,4],3); Codebook{3}(:,1:3,13)=W(:,[1,2,3],4); Codebook{3}(:,1:3,14)=W(:,[1,2,4],4); Codebook{3}(:,1:3,15)=W(:,[1,3,4],4); Codebook{3}(:,1:3,16)=W(:,[2,3,4],4);

-   -   4-Bit Rank Codebook:

Codebook{4}(:,1:4,1)=W(:,[1,2,3,4],1); Codebook{4}(:,1:4,2)=W(:,[1,2,3,4],2); Codebook{4}(:,1:4,3)=W(:,[1,2,3,4],3); Codebook{4}(:,1:4,4)=W(:,[1,2,3,4],4);

Specific numerical values of the 4-bit codebooks and the 6-bit codebooks disclosed in the above Table 15 and Table 16 may follow as (in this example ans(:,:,x) denotes the xth codeword):

1. 6-bit Codebooks

(1) 6-bit Rank 1 Codebook:

${{ans}\left( {\text{:},\text{:},1} \right)} = \begin{matrix} 0.5000 \\ 0.5000 \\ 0.5000 \\ 0.5000 \end{matrix}$ ${{ans}\left( {\text{:},\text{:},2} \right)} = \begin{matrix} 0.5000 \\ {0.0000 + {0.5000{\mathbb{i}}}} \\ {{- 0.5000} + {0.0000{\mathbb{i}}}} \\ {{- 0.0000} - {0.5000{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},3} \right)} = \begin{matrix} 0.5000 \\ {{- 0.5000} + {0.0000{\mathbb{i}}}} \\ {0.5000 - {0.0000{\mathbb{i}}}} \\ {{- 0.5000} + {0.0000{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},4} \right)} = \begin{matrix} 0.5000 \\ {{- 0.0000} - {0.5000{\mathbb{i}}}} \\ {{- 0.5000} + {0.0000{\mathbb{i}}}} \\ {0.0000 + {0.5000{\mathbb{i}}}} \end{matrix}$

${{ans}\left( {\text{:},\text{:},5} \right)} = \begin{matrix} 0.5000 \\ {0.3536 + {0.3536{\mathbb{i}}}} \\ {0.0000 + {0.5000{\mathbb{i}}}} \\ {{- 0.3536} + {0.3536{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},6} \right)} = \begin{matrix} 0.5000 \\ {{- 0.3536} + {0.3536{\mathbb{i}}}} \\ {{- 0.0000} - {0.5000{\mathbb{i}}}} \\ {0.3536 + {0.3536{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},7} \right)} = \begin{matrix} 0.5000 \\ {{- 0.3536} - {0.3536{\mathbb{i}}}} \\ {0.0000 + {0.5000{\mathbb{i}}}} \\ {0.3536 - {0.3536{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},8} \right)} = \begin{matrix} 0.5000 \\ {0.3536 - {0.3536{\mathbb{i}}}} \\ {{- 0.0000} - {0.5000{\mathbb{i}}}} \\ {{- 0.3536} - {0.3536{\mathbb{i}}}} \end{matrix}$

${{ans}\left( {\text{:},\text{:},9} \right)} = \begin{matrix} {0.4619 + {0.1913{\mathbb{i}}}} \\ {0.4455 - {0.2270{\mathbb{i}}}} \\ {{- 0.1167} + {0.4862{\mathbb{i}}}} \\ {0.2939 + {0.4045{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},10} \right)} = \begin{matrix} {{- 0.1913} + {0.4619{\mathbb{i}}}} \\ {0.4455 - {0.2270{\mathbb{i}}}} \\ {0.4862 + {0.1167{\mathbb{i}}}} \\ {{- 0.2939} - {0.4045{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},11} \right)} = \begin{matrix} {{- 0.4619} - {0.1913{\mathbb{i}}}} \\ {0.4455 - {0.2270{\mathbb{i}}}} \\ {0.1167 - {0.4862{\mathbb{i}}}} \\ {0.2939 + {0.4045{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},12} \right)} = \begin{matrix} {0.1913 - {0.4619{\mathbb{i}}}} \\ {0.4455 - {0.2270{\mathbb{i}}}} \\ {{- 0.4862} - {0.1167{\mathbb{i}}}} \\ {{- 0.2939} - {0.4045{\mathbb{i}}}} \end{matrix}$

${{ans}\left( {\text{:},\text{:},13} \right)} = \begin{matrix} {- 0.5000} \\ {0.4985 - {0.0392{\mathbb{i}}}} \\ {0.4938 - {0.0782{\mathbb{i}}}} \\ {{- 0.4862} + {0.1167{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},14} \right)} = \begin{matrix} {- 0.5000} \\ {0.0392 + {0.4985{\mathbb{i}}}} \\ {{- 0.4938} + {0.0782{\mathbb{i}}}} \\ {0.1167 + {0.4862{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},15} \right)} = \begin{matrix} {- 0.5000} \\ {{- 0.4985} + {0.0392{\mathbb{i}}}} \\ {0.4938 - {0.0782{\mathbb{i}}}} \\ {0.4862 - {0.1167{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},16} \right)} = \begin{matrix} {- 0.5000} \\ {{- 0.0392} - {0.4985{\mathbb{i}}}} \\ {{- 0.4938} + {0.0782{\mathbb{i}}}} \\ {{- 0.1167} - {0.4862{\mathbb{i}}}} \end{matrix}$

${{ans}\left( {\text{:},\text{:},17} \right)} = \begin{matrix} 0.5000 \\ {0.4619 + {0.1913{\mathbb{i}}}} \\ {0.3536 + {0.3536{\mathbb{i}}}} \\ {0.1913 + {0.4619{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},18} \right)} = \begin{matrix} 0.5000 \\ {{- 0.1913} + {0.4619{\mathbb{i}}}} \\ {{- 0.3536} - {0.3536{\mathbb{i}}}} \\ {0.4619 - {0.1913{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},19} \right)} = \begin{matrix} 0.5000` \\ {{- 0.4619} - {0.1913{\mathbb{i}}}} \\ {0.3536 + {0.3536{\mathbb{i}}}} \\ {{- 0.1913} - {0.4619{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},20} \right)} = \begin{matrix} 0.5000 \\ {0.1913 - {0.4619{\mathbb{i}}}} \\ {{- 0.3536} - {0.3536{\mathbb{i}}}} \\ {{- 0.4619} + {0.1913{\mathbb{i}}}} \end{matrix}$

${{ans}\left( {\text{:},\text{:},21} \right)} = \begin{matrix} 0.5000 \\ {0.1913 + {0.4619{\mathbb{i}}}} \\ {{- 0.3536} + {0.3536{\mathbb{i}}}} \\ {{- 0.4619} - {0.1913{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},22} \right)} = \begin{matrix} 0.5000 \\ {{- 0.4619} + {0.1913{\mathbb{i}}}} \\ {0.3536 - {0.3536{\mathbb{i}}}} \\ {{- 0.1913} + {0.4619{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},23} \right)} = \begin{matrix} 0.5000` \\ {{- 0.1913} - {0.4619{\mathbb{i}}}} \\ {{- 0.3536} + {0.3536{\mathbb{i}}}} \\ {0.4619 + {0.1913{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},24} \right)} = \begin{matrix} 0.5000 \\ {0.4619 - {0.1913{\mathbb{i}}}} \\ {0.3536 - {0.3536{\mathbb{i}}}} \\ {0.1913 - {0.4619{\mathbb{i}}}} \end{matrix}$

${{ans}\left( {\text{:},\text{:},25} \right)} = \begin{matrix} {0.2437 + {0.4837{\mathbb{i}}}} \\ {0.5740 - {0.3102{\mathbb{i}}}} \\ {0.2823 - {0.3783{\mathbb{i}}}} \\ {0.2062 + {0.1248{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},26} \right)} = \begin{matrix} {0.4258 + {0.0076{\mathbb{i}}}} \\ {0.0596 + {0.3222{\mathbb{i}}}} \\ {{- 0.4186} + {0.1136{\mathbb{i}}}} \\ {0.1513 + {0.7073{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},27} \right)} = \begin{matrix} {0.5210 - {0.4265{\mathbb{i}}}} \\ {0.2922 + {0.0454{\mathbb{i}}}} \\ {0.3628 + {0.1522{\mathbb{i}}}} \\ {{- 0.5498} - {0.0464{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},28} \right)} = \begin{matrix} {{- 0.2364} - {0.1268{\mathbb{i}}}} \\ {0.6147 + {0.0409{\mathbb{i}}}} \\ {{- 0.6500} - {0.1085{\mathbb{i}}}} \\ {{- 0.2340} - {0.2440{\mathbb{i}}}} \end{matrix}$

${{ans}\left( {\text{:},\text{:},29} \right)} = \begin{matrix} {0.0591 - {0.4953{\mathbb{i}}}} \\ {0.5545 + {0.3908{\mathbb{i}}}} \\ {0.2593 - {0.0234{\mathbb{i}}}} \\ {0.3300 + {0.3380{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},30} \right)} = \begin{matrix} {0.0742 + {0.2714{\mathbb{i}}}} \\ {{- 0.0591} - {0.0963{\mathbb{i}}}} \\ {{- 0.5800} - {0.1645{\mathbb{i}}}} \\ {0.4619 + {0.5755{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},31} \right)} = \begin{matrix} {0.3370 + {0.2000{\mathbb{i}}}} \\ {0.2158 + {0.4089{\mathbb{i}}}} \\ {{- 0.2363} - {0.5965{\mathbb{i}}}} \\ {0.0175 - {0.4698{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},32} \right)} = \begin{matrix} {{- 0.1041} - {0.7125{\mathbb{i}}}} \\ {{- 0.5589} + {0.0299{\mathbb{i}}}} \\ {{- 0.2149} - {0.3331{\mathbb{i}}}} \\ {0.0748 - {0.0748{\mathbb{i}}}} \end{matrix}$

${{ans}\left( {\text{:},\text{:},33} \right)} = \begin{matrix} {0.5019 - {0.2171{\mathbb{i}}}} \\ {0.0570 + {0.3132{\mathbb{i}}}} \\ {0.2569 - {0.1797{\mathbb{i}}}} \\ {0.6949 + {0.1358{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},34} \right)} = \begin{matrix} {0.2121 - {0.4391{\mathbb{i}}}} \\ {0.3637 + {0.1907{\mathbb{i}}}} \\ {0.0539 - {0.3157{\mathbb{i}}}} \\ {{- 0.5835} + {0.3879{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},35} \right)} = \begin{matrix} {0.4649 + {0.1545{\mathbb{i}}}} \\ {0.5971 - {0.5122{\mathbb{i}}}} \\ {{- 0.0025} + {0.3671{\mathbb{i}}}} \\ {0.0058 - {0.0797{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},36} \right)} = \begin{matrix} {{- 0.4709} + {0.0383{\mathbb{i}}}} \\ {0.3285 + {0.0569{\mathbb{i}}}} \\ {0.8042 + {0.1327{\mathbb{i}}}} \\ {0.0341 + {0.0124{\mathbb{i}}}} \end{matrix}$

${{ans}\left( {\text{:},\text{:},37} \right)} = \begin{matrix} {0.2199 + {0.3199{\mathbb{i}}}} \\ {0.5983 + {0.1264{\mathbb{i}}}} \\ {{- 0.5576} - {0.2903{\mathbb{i}}}} \\ {{- 0.0940} - {0.2672{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},38} \right)} = \begin{matrix} {0.6305 - {0.4345{\mathbb{i}}}} \\ {{- 0.0683} + {0.2985{\mathbb{i}}}} \\ {{- 0.0495} - {0.3235{\mathbb{i}}}} \\ {{- 0.0481} + {0.4588{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},39} \right)} = \begin{matrix} {{- 0.0525} + {0.2357{\mathbb{i}}}} \\ {0.5833 + {0.2614{\mathbb{i}}}} \\ {0.3552 + {0.3750{\mathbb{i}}}} \\ {0.0118 + {0.5160{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},40} \right)} = \begin{matrix} {{- 0.3110} - {0.3286{\mathbb{i}}}} \\ {0.2969 - {0.1888{\mathbb{i}}}} \\ {0.2888 - {0.3841{\mathbb{i}}}} \\ {{- 0.6633} - {0.0260{\mathbb{i}}}} \end{matrix}$

${{ans}\left( {\text{:},\text{:},41} \right)} = \begin{matrix} {0.4534 + {0.1075{\mathbb{i}}}} \\ {{- 0.0163} + {0.7556{\mathbb{i}}}} \\ {0.0681 + {0.3202{\mathbb{i}}}} \\ {0.1259 + {0.2977{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},42} \right)} = \begin{matrix} {0.2632 + {0.4348{\mathbb{i}}}} \\ {0.3698 - {0.0222{\mathbb{i}}}} \\ {{- 0.4066} - {0.6110{\mathbb{i}}}} \\ {0.0347 + {0.2542{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},43} \right)} = \begin{matrix} {0.1838 + {0.6429{\mathbb{i}}}} \\ {0.0055 - {0.3112{\mathbb{i}}}} \\ {0.3882 + {0.2198{\mathbb{i}}}} \\ {0.4482 - {0.2369{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},44} \right)} = \begin{matrix} {0.2307 + {0.1558{\mathbb{i}}}} \\ {0.3467 - {0.2728{\mathbb{i}}}} \\ {0.1705 + {0.3551{\mathbb{i}}}} \\ {{- 0.7319} + {0.1924{\mathbb{i}}}} \end{matrix}$

${{ans}\left( {\text{:},\text{:},45} \right)} = \begin{matrix} {0.0534 - {0.5633{\mathbb{i}}}} \\ {{- 0.1472} - {0.1184{\mathbb{i}}}} \\ {{- 0.7445} + {0.0533{\mathbb{i}}}} \\ {0.2881 - {0.0638{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},46} \right)} = \begin{matrix} {{- 0.4493} + {0.3172{\mathbb{i}}}} \\ {{- 0.5422} + {0.3203{\mathbb{i}}}} \\ {{- 0.1168} - {0.4073{\mathbb{i}}}} \\ {0.3438 + {0.0562{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},47} \right)} = \begin{matrix} {{- 0.3346} + {0.1479{\mathbb{i}}}} \\ {0.2302 - {0.4770{\mathbb{i}}}} \\ {{- 0.1288} - {0.3642{\mathbb{i}}}} \\ {{- 0.1358} - {0.6465{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},48} \right)} = \begin{matrix} {{- 0.4868} - {0.0811{\mathbb{i}}}} \\ {0.4913 + {0.2140{\mathbb{i}}}} \\ {{- 0.2957} - {0.1634{\mathbb{i}}}} \\ {{- 0.3579} + {0.4766{\mathbb{i}}}} \end{matrix}$

${{ans}\left( {\text{:},\text{:},49} \right)} = \begin{matrix} {0.4192 + {0.3317{\mathbb{i}}}} \\ {{- 0.2340} + {0.0529{\mathbb{i}}}} \\ {0.0491 - {0.2199{\mathbb{i}}}} \\ {{- 0.7746} + {0.0772{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},50} \right)} = \begin{matrix} {0.6462 - {0.3049{\mathbb{i}}}} \\ {{- 0.0388} - {0.4789{\mathbb{i}}}} \\ {{- 0.2962} + {0.1142{\mathbb{i}}}} \\ {0.1089 - {0.3821{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},51} \right)} = \begin{matrix} {0.1765 - {0.0924{\mathbb{i}}}} \\ {0.6212 - {0.0425{\mathbb{i}}}} \\ {0.5130 + {0.5000{\mathbb{i}}}} \\ {{- 0.2395} + {0.0459{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},52} \right)} = \begin{matrix} {0.3680 + {0.1691{\mathbb{i}}}} \\ {{- 0.5223} + {0.2262{\mathbb{i}}}} \\ {0.5366 + {0.2175{\mathbb{i}}}} \\ {0.4196 + {0.0255{\mathbb{i}}}} \end{matrix}$

${{ans}\left( {\text{:},\text{:},53} \right)} = \begin{matrix} {0.0848 + {0.2248{\mathbb{i}}}} \\ {{- 0.2010} + {0.7357{\mathbb{i}}}} \\ {0.3862 + {0.2270{\mathbb{i}}}} \\ {{- 0.3114} + {0.2509{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},54} \right)} = \begin{matrix} {0.5883 + {0.1906{\mathbb{i}}}} \\ {{- 0.0481} - {0.2343{\mathbb{i}}}} \\ {0.2323 + {0.5375{\mathbb{i}}}} \\ {0.4661 + {0.0138{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},55} \right)} = \begin{matrix} {{- 0.4550} - {0.1147{\mathbb{i}}}} \\ {{- 0.3923} + {0.0511{\mathbb{i}}}} \\ {0.1654 - {0.1078{\mathbb{i}}}} \\ {0.6358 + {0.4245{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},56} \right)} = \begin{matrix} {{- 0.5202} + {0.2628{\mathbb{i}}}} \\ {0.3107 - {0.3288{\mathbb{i}}}} \\ {0.6089 + {0.2161{\mathbb{i}}}} \\ {{- 0.1325} - {0.1438{\mathbb{i}}}} \end{matrix}$

${{ans}\left( {\text{:},\text{:},57} \right)} = \begin{matrix} {0.3968 - {0.0671{\mathbb{i}}}} \\ {{- 0.8289} - {0.0667{\mathbb{i}}}} \\ {0.1770 - {0.1687{\mathbb{i}}}} \\ {{- 0.2814} + {0.0866{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},58} \right)} = \begin{matrix} {0.3110 + {0.1356{\mathbb{i}}}} \\ {{- 0.2015} + {0.0416{\mathbb{i}}}} \\ {{- 0.3325} - {0.1370{\mathbb{i}}}} \\ {0.7347 - {0.4165{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},59} \right)} = \begin{matrix} {{- 0.2168} - {0.4540{\mathbb{i}}}} \\ {0.1102 - {0.2871{\mathbb{i}}}} \\ {0.0567 - {0.7634{\mathbb{i}}}} \\ {{- 0.0390} - {0.2544{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},60} \right)} = \begin{matrix} {{- 0.6008} - {0.3299{\mathbb{i}}}} \\ {{- 0.4011} - {0.1031{\mathbb{i}}}} \\ {0.2785 + {0.3837{\mathbb{i}}}} \\ {0.3519 - {0.1000{\mathbb{i}}}} \end{matrix}$

${{ans}\left( {\text{:},\text{:},61} \right)} = \begin{matrix} {0.0743 - {0.7846{\mathbb{i}}}} \\ {0.1307 - {0.3436{\mathbb{i}}}} \\ {{- 0.3387} - {0.2385{\mathbb{i}}}} \\ {{- 0.0962} + {0.2508{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},62} \right)} = \begin{matrix} {{- 0.4063} + {0.3394{\mathbb{i}}}} \\ {0.2466 + {0.0922{\mathbb{i}}}} \\ {{- 0.4080} - {0.5055{\mathbb{i}}}} \\ {{- 0.4767} - {0.0347{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},63} \right)} = \begin{matrix} {{- 0.2621} + {0.0798{\mathbb{i}}}} \\ {0.4783 - {0.5634{\mathbb{i}}}} \\ {0.5212 + {0.1951{\mathbb{i}}}} \\ {{- 0.2418} + {0.1026{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},64} \right)} = \begin{matrix} {0.1421 + {0.0583{\mathbb{i}}}} \\ {{- 0.0799} - {0.4928{\mathbb{i}}}} \\ {{- 0.2917} + {0.1078{\mathbb{i}}}} \\ {{- 0.0231} - {0.7937{\mathbb{i}}}} \end{matrix}$

(2) 6-Bit Rank 2 Codebook:

${{ans}\left( {\text{:},\text{:},1} \right)} = \begin{matrix} 0.5000 & 0.5000 \\ 0.5000 & {0.0000 + {0.5000{\mathbb{i}}}} \\ 0.5000 & {{- 0.5000} + {0.0000{\mathbb{i}}}} \\ 0.5000 & {{- 0.0000} - {0.5000{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},2} \right)} = \begin{matrix} 0.5000 & 0.5000 \\ 0.5000 & {{- 0.5000} + {0.0000{\mathbb{i}}}} \\ 0.5000 & {0.5000 - {0.0000{\mathbb{i}}}} \\ 0.5000 & {{- 0.5000} + {0.0000{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},3} \right)} = \begin{matrix} 0.5000 & 0.5000 \\ 0.5000 & {{- 0.0000} - {0.5000{\mathbb{i}}}} \\ 0.5000 & {{- 0.5000} + {0.0000{\mathbb{i}}}} \\ 0.5000 & {0.0000 + {0.5000{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},4} \right)} = \begin{matrix} 0.5000 & 0.5000 \\ {0.0000 + {0.5000{\mathbb{i}}}} & {{- 0.5000} + {0.0000{\mathbb{i}}}} \\ {{- 0.5000} + {0.0000{\mathbb{i}}}} & {0.5000 - {0.0000{\mathbb{i}}}} \\ {{- 0.0000} - {0.5000{\mathbb{i}}}} & {{- 0.5000} + {0.0000{\mathbb{i}}}} \end{matrix}$

${{ans}\left( {\text{:},\text{:},5} \right)} = \begin{matrix} 0.5000 & 0.5000 \\ {0.0000 + {0.5000{\mathbb{i}}}} & {{- 0.0000} - {0.5000{\mathbb{i}}}} \\ {{- 0.5000} + {0.0000{\mathbb{i}}}} & {{- 0.5000} + {0.0000{\mathbb{i}}}} \\ {{- 0.0000} - {0.5000{\mathbb{i}}}} & {0.0000 + {0.5000{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},6} \right)} = \begin{matrix} 0.5000 & 0.5000 \\ {{- 0.5000} + {0.0000{\mathbb{i}}}} & {{- 0.0000} - {0.5000{\mathbb{i}}}} \\ {0.5000 - {0.0000{\mathbb{i}}}} & {{- 0.5000} + {0.0000{\mathbb{i}}}} \\ {{- 0.5000} + {0.0000{\mathbb{i}}}} & {0.0000 + {0.5000{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},7} \right)} = \begin{matrix} 0.5000 & 0.5000 \\ {0.3536 + {0.3536{\mathbb{i}}}} & {{- 0.3536} + {0.3536{\mathbb{i}}}} \\ {0.0000 + {0.5000{\mathbb{i}}}} & {{- 0.0000} - {0.5000{\mathbb{i}}}} \\ {{- 0.3536} + {0.3536{\mathbb{i}}}} & {0.3536 + {0.3536{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},8} \right)} = \begin{matrix} 0.5000 & 0.5000 \\ {0.3536 + {0.3536{\mathbb{i}}}} & {{- 0.3536} - {0.3536{\mathbb{i}}}} \\ {0.0000 + {0.5000{\mathbb{i}}}} & {0.0000 + {0.5000{\mathbb{i}}}} \\ {{- 0.3536} + {0.3536{\mathbb{i}}}} & {0.3536 - {0.3536{\mathbb{i}}}} \end{matrix}$

${{ans}\left( {\text{:},\text{:},9} \right)} = \begin{matrix} 0.5000 & 0.5000 \\ {0.3536 + {0.3536{\mathbb{i}}}} & {0.3536 - {0.3536{\mathbb{i}}}} \\ {0.0000 + {0.5000{\mathbb{i}}}} & {{- 0.0000} - {0.5000{\mathbb{i}}}} \\ {{- 0.3536} + {0.3536{\mathbb{i}}}} & {{- 0.3536} - {0.3536{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},10} \right)} = \begin{matrix} 0.5000 & 0.5000 \\ {{- 0.3536} + {0.3536{\mathbb{i}}}} & {{- 0.3536} - {0.3536{\mathbb{i}}}} \\ {{- 0.0000} - {0.5000{\mathbb{i}}}} & {0.0000 + {0.5000{\mathbb{i}}}} \\ {0.3536 + {0.3536{\mathbb{i}}}} & {0.3536 - {0.3536{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},11} \right)} = \begin{matrix} 0.5000 & 0.5000 \\ {{- 0.3536} + {0.3536{\mathbb{i}}}} & {0.3536 - {0.3536{\mathbb{i}}}} \\ {{- 0.0000} - {0.5000{\mathbb{i}}}} & {{- 0.0000} - {0.5000{\mathbb{i}}}} \\ {0.3536 + {0.3536{\mathbb{i}}}} & {{- 0.3536} - {0.3536{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},12} \right)} = \begin{matrix} 0.5000 & 0.5000 \\ {{- 0.3536} - {0.3536{\mathbb{i}}}} & {0.3536 - {0.3536{\mathbb{i}}}} \\ {0.0000 + {0.5000{\mathbb{i}}}} & {{- 0.0000} - {0.5000{\mathbb{i}}}} \\ {0.3536 - {0.3536{\mathbb{i}}}} & {{- 0.3536} - {0.3536{\mathbb{i}}}} \end{matrix}$

${{ans}\left( {\text{:},\text{:},13} \right)} = \begin{matrix} {0.4619 + {0.1913{\mathbb{i}}}} & {0.1913 - {0.4619{\mathbb{i}}}} \\ {0.4455 - {0.2270{\mathbb{i}}}} & {0.4455 - {0.2270{\mathbb{i}}}} \\ {{- 0.1167} + {0.4862{\mathbb{i}}}} & {{- 0.4862} - {0.1167{\mathbb{i}}}} \\ {0.2939 + {0.4045{\mathbb{i}}}} & {{- 0.2939} - {0.4045{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},14} \right)} = \begin{matrix} {{- 0.1913} + {0.4619{\mathbb{i}}}} & {{- 0.4619} - {0.1913{\mathbb{i}}}} \\ {0.4455 - {0.2270{\mathbb{i}}}} & {0.4455 - {0.2270{\mathbb{i}}}} \\ {0.4862 + {0.1167{\mathbb{i}}}} & {0.1167 - {0.4862{\mathbb{i}}}} \\ {{- 0.2939} - {0.4045{\mathbb{i}}}} & {0.2939 + {0.4045{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},15} \right)} = \begin{matrix} {- 0.5000} & {- 0.5000} \\ {0.4985 - {0.0392{\mathbb{i}}}} & {0.0392 + {0.4985{\mathbb{i}}}} \\ {0.4938 - {0.0782{\mathbb{i}}}} & {{- 0.4938} + {0.0782{\mathbb{i}}}} \\ {{- 0.4862} + {0.1167{\mathbb{i}}}} & {0.1167 + {0.4862{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},16} \right)} = \begin{matrix} {- 0.5000} & {- 0.5000} \\ {0.4985 - {0.0392{\mathbb{i}}}} & {{- 0.0392} - {0.4985{\mathbb{i}}}} \\ {0.4938 - {0.0782{\mathbb{i}}}} & {{- 0.4938} + {0.0782{\mathbb{i}}}} \\ {{- 0.4862} + {0.1167{\mathbb{i}}}} & {{- 0.1167} - {0.4862{\mathbb{i}}}} \end{matrix}$

${{ans}\left( {\text{:},\text{:},17} \right)} = \begin{matrix} 0.5000 & 0.5000 \\ {0.4619 + {0.1913{\mathbb{i}}}} & {{- 0.1913} + {0.4619{\mathbb{i}}}} \\ {0.3536 + {0.3536{\mathbb{i}}}} & {{- 0.3536} - {0.3536{\mathbb{i}}}} \\ {0.1913 + {0.4619{\mathbb{i}}}} & {0.4619 - {0.1913{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},18} \right)} = \begin{matrix} 0.5000 & 0.5000 \\ {0.1913 + {0.4619{\mathbb{i}}}} & {{- 0.1913} - {0.4619{\mathbb{i}}}} \\ {{- 0.3536} + {0.3536{\mathbb{i}}}} & {{- 0.3536} + {0.3536{\mathbb{i}}}} \\ {{- 0.4619} - {0.1913{\mathbb{i}}}} & {0.4619 + {0.1913{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},19} \right)} = \begin{matrix} 0.5000 & 0.5000 \\ {{- 0.4619} + {0.1913{\mathbb{i}}}} & {{- 0.1913} - {0.4619{\mathbb{i}}}} \\ {0.3536 - {0.3536{\mathbb{i}}}} & {{- 0.3536} + {0.3536{\mathbb{i}}}} \\ {{- 0.1913} + {0.4619{\mathbb{i}}}} & {0.4619 + {0.1913{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},20} \right)} = \begin{matrix} 0.5000 & 0.5000 \\ {{- 0.4619} + {0.1913{\mathbb{i}}}} & {0.4619 - {0.1913{\mathbb{i}}}} \\ {0.3536 - {0.3536{\mathbb{i}}}} & {0.3536 - {0.3536{\mathbb{i}}}} \\ {{- 0.1913} + {0.4619{\mathbb{i}}}} & {0.1913 - {0.4619{\mathbb{i}}}} \end{matrix}$

${{ans}\left( {\text{:},\text{:},21} \right)} = \begin{matrix} {0.2437 + {0.4837{\mathbb{i}}}} & {0.4258 + {0.0076{\mathbb{i}}}} \\ {0.5740 - {0.3102{\mathbb{i}}}} & {0.0596 + {0.3222{\mathbb{i}}}} \\ {0.2823 - {0.3783{\mathbb{i}}}} & {{- 0.4186} + {0.1136{\mathbb{i}}}} \\ {0.2062 + {0.1248{\mathbb{i}}}} & {0.1513 + {0.7073{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},22} \right)} = \begin{matrix} {0.2437 + {0.4837{\mathbb{i}}}} & {0.5210 - {0.4265{\mathbb{i}}}} \\ {0.5740 - {0.3102{\mathbb{i}}}} & {0.2922 + {0.0454{\mathbb{i}}}} \\ {0.2823 - {0.3783{\mathbb{i}}}} & {0.3826 + {0.1522{\mathbb{i}}}} \\ {0.2062 + {0.1248{\mathbb{i}}}} & {{- 0.5498} - {0.0464{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},23} \right)} = \begin{matrix} {0.2437 + {0.4837{\mathbb{i}}}} & {{- 0.2364} - {0.1268{\mathbb{i}}}} \\ {0.5740 - {0.3102{\mathbb{i}}}} & {0.6147 + {0.0409{\mathbb{i}}}} \\ {0.2823 - {0.3783{\mathbb{i}}}} & {{- 0.6500} - {0.1085{\mathbb{i}}}} \\ {0.2062 + {0.1248{\mathbb{i}}}} & {{- 0.2340} - {0.2440{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},24} \right)} = \begin{matrix} {0.4258 + {0.0076{\mathbb{i}}}} & {0.5210 - {0.4265{\mathbb{i}}}} \\ {0.0596 + {0.3222{\mathbb{i}}}} & {0.2922 + {0.0454{\mathbb{i}}}} \\ {{- 0.4186} + {0.1136{\mathbb{i}}}} & {0.3628 + {0.1522{\mathbb{i}}}} \\ {0.1513 + {0.7073{\mathbb{i}}}} & {{- 0.5498} - {0.0464{\mathbb{i}}}} \end{matrix}$

${{ans}\left( {\text{:},\text{:},25} \right)} = \begin{matrix} {0.4258 + {0.0076{\mathbb{i}}}} & {{- 0.2364} - {0.1268{\mathbb{i}}}} \\ {0.0596 + {0.3222{\mathbb{i}}}} & {0.6147 + {0.0409{\mathbb{i}}}} \\ {{- 0.4186} + {0.1136{\mathbb{i}}}} & {{- 0.6500} - {0.1085{\mathbb{i}}}} \\ {0.1513 + {0.7073{\mathbb{i}}}} & {{- 0.2340} - {0.2440{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},26} \right)} = \begin{matrix} {0.5210 - {0.4265{\mathbb{i}}}} & {{- 0.2364} - {0.1268{\mathbb{i}}}} \\ {0.2922 + {0.0454{\mathbb{i}}}} & {0.6147 + {0.0409{\mathbb{i}}}} \\ {0.3628 + {0.1522{\mathbb{i}}}} & {{- 0.6500} - {0.1085{\mathbb{i}}}} \\ {{- 0.5498} - {0.0464{\mathbb{i}}}} & {{- 0.2340} - {0.2440{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},27} \right)} = \begin{matrix} {0.0591 - {0.4953{\mathbb{i}}}} & {0.0742 + {0.2714{\mathbb{i}}}} \\ {0.5545 + {0.3908{\mathbb{i}}}} & {{- 0.0591} - {0.0963{\mathbb{i}}}} \\ {0.2593 - {0.0234{\mathbb{i}}}} & {{- 0.5800} - {0.1645{\mathbb{i}}}} \\ {0.3300 + {0.3380{\mathbb{i}}}} & {0.4619 + {0.5755{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},28} \right)} = \begin{matrix} {0.0591 - {0.4953{\mathbb{i}}}} & {0.3370 + {0.2000{\mathbb{i}}}} \\ {0.5545 + {0.3908{\mathbb{i}}}} & {0.2158 + {0.4089{\mathbb{i}}}} \\ {0.2593 - {0.0234{\mathbb{i}}}} & {{- 0.2363} - {0.5965{\mathbb{i}}}} \\ {0.3300 + {0.3380{\mathbb{i}}}} & {0.0175 - {0.4698{\mathbb{i}}}} \end{matrix}$

${{ans}\left( {\text{:},\text{:},29} \right)} = \begin{matrix} {0.0591 - {0.4953{\mathbb{i}}}} & {{- 0.1041} - {0.7125{\mathbb{i}}}} \\ {0.5545 + {0.3908{\mathbb{i}}}} & {{- 0.5589} + {0.0299{\mathbb{i}}}} \\ {0.2593 - {0.0234{\mathbb{i}}}} & {{- 0.2149} - {0.3331{\mathbb{i}}}} \\ {0.3300 + {0.3380{\mathbb{i}}}} & {0.0748 - {0.0748{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},30} \right)} = \begin{matrix} {0.0742 + {0.2714{\mathbb{i}}}} & {0.3370 + {0.2000{\mathbb{i}}}} \\ {{- 0.0591} - {0.0963{\mathbb{i}}}} & {0.2158 + {0.4089{\mathbb{i}}}} \\ {{- 0.5800} - {0.1645{\mathbb{i}}}} & {{- 0.2363} - {0.5965{\mathbb{i}}}} \\ {0.4619 + {0.5755{\mathbb{i}}}} & {0.0175 - {0.4698{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},31} \right)} = \begin{matrix} {0.0742 + {0.2714{\mathbb{i}}}} & {{- 0.1041} - {0.7125{\mathbb{i}}}} \\ {{- 0.0591} - {0.0963{\mathbb{i}}}} & {{- 0.5589} + {0.0299{\mathbb{i}}}} \\ {{- 0.5800} - {0.1645{\mathbb{i}}}} & {{- 0.2149} - {0.3331{\mathbb{i}}}} \\ {0.4619 + {0.5755{\mathbb{i}}}} & {{0.0748 - {0.0748{\mathbb{i}}}}\;} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},32} \right)} = \begin{matrix} {0.3370 + {0.2000{\mathbb{i}}}} & {{- 0.1041} - {0.7125{\mathbb{i}}}} \\ {0.2158 + {0.4089{\mathbb{i}}}} & {{- 0.5589} + {0.0299{\mathbb{i}}}} \\ {{- 0.2363} - {0.5965{\mathbb{i}}}} & {{- 0.2149} - {0.3331{\mathbb{i}}}} \\ {0.0175 - {0.4698{\mathbb{i}}}} & {0.0748 - {0.0748{\mathbb{i}}}} \end{matrix}$

${{ans}\left( {\text{:},\text{:},33} \right)} = \begin{matrix} {0.5019 - {0.2171{\mathbb{i}}}} & {0.2121 - {0.4391{\mathbb{i}}}} \\ {0.0570 + {0.3132{\mathbb{i}}}} & {0.3637 + {0.1907{\mathbb{i}}}} \\ {0.2569 - {0.1797{\mathbb{i}}}} & {0.0539 - {0.3157{\mathbb{i}}}} \\ {0.6949 + {0.1358{\mathbb{i}}}} & {{- 0.5835} + {0.3879{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},34} \right)} = \begin{matrix} {0.5019 - {0.2171{\mathbb{i}}}} & {0.4649 + {0.1545{\mathbb{i}}}} \\ {0.0570 + {0.3132{\mathbb{i}}}} & {0.5971 - {0.5122{\mathbb{i}}}} \\ {0.2569 - {0.1797{\mathbb{i}}}} & {{- 0.0025} + {0.3671{\mathbb{i}}}} \\ {0.6949 + {0.1358{\mathbb{i}}}} & {0.0058 - {0.0797{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},35} \right)} = \begin{matrix} {0.2121 - {0.4391{\mathbb{i}}}} & {0.4649 + {0.1545{\mathbb{i}}}} \\ {0.3637 + {0.1907{\mathbb{i}}}} & {0.5971 - {0.5122{\mathbb{i}}}} \\ {0.0539 - {0.3157{\mathbb{i}}}} & {{- 0.0025} + {0.3671{\mathbb{i}}}} \\ {{- 0.5835} + {0.3879{\mathbb{i}}}} & {0.0058 - {0.0797{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},36} \right)} = \begin{matrix} {0.2121 - {0.4391{\mathbb{i}}}} & {{- 0.4709} + {0.0383{\mathbb{i}}}} \\ {0.3637 + {0.1907{\mathbb{i}}}} & {0.3285 + {0.0569{\mathbb{i}}}} \\ {0.0539 - {0.3157{\mathbb{i}}}} & {0.8042 + {0.1327{\mathbb{i}}}} \\ {{- 0.5835} + {0.3879{\mathbb{i}}}} & {0.0341 + {0.0124{\mathbb{i}}}} \end{matrix}$

${{ans}\left( {\text{:},\text{:},37} \right)} = \begin{matrix} {0.2199 + {0.3199{\mathbb{i}}}} & {{- 0.0525} + {0.2357{\mathbb{i}}}} \\ {0.5983 + {0.1264{\mathbb{i}}}} & {0.5833 + {0.2614{\mathbb{i}}}} \\ {{- 0.5576} - {0.2903{\mathbb{i}}}} & {0.3552 + {0.3750{\mathbb{i}}}} \\ {{- 0.0940} - {0.2672{\mathbb{i}}}} & {0.0118 + {0.5160{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},38} \right)} = \begin{matrix} {0.6305 - {0.4345{\mathbb{i}}}} & {{- 0.3110} - {0.3286{\mathbb{i}}}} \\ {{- 0.0683} + {0.2985{\mathbb{i}}}} & {0.2969 - {0.1888{\mathbb{i}}}} \\ {{- 0.0495} - {0.3235{\mathbb{i}}}} & {0.2888 - {0.3841{\mathbb{i}}}} \\ {{- 0.0481} + {0.4588{\mathbb{i}}}} & {{- 0.6633} - {0.0260{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},39} \right)} = \begin{matrix} {{- 0.0525} + {0.2357{\mathbb{i}}}} & {{- 0.3110} - {0.3286{\mathbb{i}}}} \\ {0.5833 + {0.2614{\mathbb{i}}}} & {0.2969 - {0.1888{\mathbb{i}}}} \\ {0.3552 + {0.3750{\mathbb{i}}}} & {0.2888 - {0.3841{\mathbb{i}}}} \\ {0.0118 + {0.5160{\mathbb{i}}}} & {{- 0.6633} - {0.0260{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},40} \right)} = \begin{matrix} {0.4534 + {0.1075{\mathbb{i}}}} & {0.2632 + {0.4348{\mathbb{i}}}} \\ {{- 0.0163} + {0.7556{\mathbb{i}}}} & {0.3698 - {0.0222{\mathbb{i}}}} \\ {0.0681 + {0.3202{\mathbb{i}}}} & {{- 0.4066} - {0.6110{\mathbb{i}}}} \\ {0.1259 + {0.2977{\mathbb{i}}}} & {0.0347 + {0.2542{\mathbb{i}}}} \end{matrix}$

${{ans}\left( {\text{:},\text{:},41} \right)} = \begin{matrix} {0.4534 + {0.1075{\mathbb{i}}}} & {0.1838 + {0.6429{\mathbb{i}}}} \\ {{- 0.0163} + {0.7556{\mathbb{i}}}} & {0.0055 - {0.3112{\mathbb{i}}}} \\ {0.0681 + {0.3202{\mathbb{i}}}} & {0.3882 + {0.2198{\mathbb{i}}}} \\ {0.1259 + {0.2977{\mathbb{i}}}} & {0.4482 - {0.2369{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},42} \right)} = \begin{matrix} {0.4534 + {0.1075{\mathbb{i}}}} & {0.2307 + {0.1558{\mathbb{i}}}} \\ {{- 0.0163} + {0.7556{\mathbb{i}}}} & {0.3467 - {0.2728{\mathbb{i}}}} \\ {0.0681 + {0.3202{\mathbb{i}}}} & {0.1705 + {0.3551{\mathbb{i}}}} \\ {0.1259 + {0.2977{\mathbb{i}}}} & {{- 0.7319} + {0.1924{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},43} \right)} = \begin{matrix} {0.2632 + {0.4348{\mathbb{i}}}} & {0.1838 + {0.6429{\mathbb{i}}}} \\ {0.3698 - {0.0222{\mathbb{i}}}} & {0.0055 - {0.3112{\mathbb{i}}}} \\ {{- 0.4066} - {0.6110{\mathbb{i}}}} & {0.3882 + {0.2198{\mathbb{i}}}} \\ {0.0347 + {0.2542{\mathbb{i}}}} & {0.4482 - {0.2369{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},44} \right)} = \begin{matrix} {0.2632 + {0.4348{\mathbb{i}}}} & {0.2307 + {0.1558{\mathbb{i}}}} \\ {0.3698 - {0.0222{\mathbb{i}}}} & {0.3467 - {0.2728{\mathbb{i}}}} \\ {{- 0.4066} - {0.6110{\mathbb{i}}}} & {0.1705 + {0.3551{\mathbb{i}}}} \\ {0.0347 + {0.2542{\mathbb{i}}}} & {0.7319 + {0.1924{\mathbb{i}}}} \end{matrix}$

${{ans}\left( {\text{:},\text{:},45} \right)} = \begin{matrix} {0.1838 + {0.6429{\mathbb{i}}}} & {0.2307 + {0.1558{\mathbb{i}}}} \\ {0.0055 - {0.3112{\mathbb{i}}}} & {0.3467 - {0.2728{\mathbb{i}}}} \\ {0.3882 + {0.2198{\mathbb{i}}}} & {0.1705 + {03551{\mathbb{i}}}} \\ {0.4482 - {0.2369{\mathbb{i}}}} & {{- 0.7319} + {0.1924{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},46} \right)} = \begin{matrix} {0.0534 - {0.5633{\mathbb{i}}}} & {{- 0.4493} + {0.3172{\mathbb{i}}}} \\ {{- 0.1472} - {0.1184{\mathbb{i}}}} & {{- 0.5422} + {0.3203{\mathbb{i}}}} \\ {{- 0.7445} + {0.0533{\mathbb{i}}}} & {{- 0.1168} - {0.4073{\mathbb{i}}}} \\ {0.2281 - {0.0638{\mathbb{i}}}} & {0.3438 + {0.0562{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},47} \right)} = \begin{matrix} {0.0534 - {0.5633{\mathbb{i}}}} & {{- 0.3346} + {0.1479{\mathbb{i}}}} \\ {{- 0.1472} - {0.1184{\mathbb{i}}}} & {0.2302 - {0.4770{\mathbb{i}}}} \\ {{- 0.7445} + {0.0533{\mathbb{i}}}} & {{- 0.1288} - {0.3642{\mathbb{i}}}} \\ {0.2881 - {0.0638{\mathbb{i}}}} & {{- 0.1358} - {0.6465{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},48} \right)} = \begin{matrix} {{- 0.4493} + {0.3172{\mathbb{i}}}} & {{- 0.3346} + {0.1479{\mathbb{i}}}} \\ {{- 0.5422} + {0.3203{\mathbb{i}}}} & {0.2302 - {0.4770{\mathbb{i}}}} \\ {{- 0.1168} - {0.4073{\mathbb{i}}}} & {{- 0.1288} - {0.3642{\mathbb{i}}}} \\ {0.3438 + {0.0562{\mathbb{i}}}} & {{- 0.1358} - {0.6465{\mathbb{i}}}} \end{matrix}$

${{ans}\left( {\text{:},\text{:},49} \right)} = \begin{matrix} {0.4192 + {0.3317{\mathbb{i}}}} & {0.6462 - {0.3049{\mathbb{i}}}} \\ {{- 0.2340} + {0.0529{\mathbb{i}}}} & {{- 0.0388} - {0.4789{\mathbb{i}}}} \\ {0.0491 - {0.2199{\mathbb{i}}}} & {{- 0.2962} + {0.1142{\mathbb{i}}}} \\ {{- 0.7746} + {0.0772{\mathbb{i}}}} & {0.1089 - {0.3821{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},50} \right)} = \begin{matrix} {0.4192 + {0.3317{\mathbb{i}}}} & {0.3680 + {0.1691{\mathbb{i}}}} \\ {{- 0.2340} + {0.0529{\mathbb{i}}}} & {{- 0.5223} + {0.2262{\mathbb{i}}}} \\ {0.0491 - {0.2199{\mathbb{i}}}} & {0.5366 + {0.2175{\mathbb{i}}}} \\ {{- 0.7746} + {0.0772{\mathbb{i}}}} & {0.4196 + {0.0255{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},51} \right)} = \begin{matrix} {0.6462 - {0.3049{\mathbb{i}}}} & {0.3680 + {0.1691{\mathbb{i}}}} \\ {{- 0.0388} - {0.4789{\mathbb{i}}}} & {{- 0.5223} + {0.2262{\mathbb{i}}}} \\ {{- 0.2962} + {0.1142{\mathbb{i}}}} & {0.5366 + {0.2175{\mathbb{i}}}} \\ {0.1089 - {0.3821{\mathbb{i}}}} & {0.4196 + {0.0255{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},52} \right)} = \begin{matrix} {0.1765 - {0.0924{\mathbb{i}}}} & {0.3680 + {0.1691{\mathbb{i}}}} \\ {0.6212 - {0.0425{\mathbb{i}}}} & {{- 0.5223} + {0.2262{\mathbb{i}}}} \\ {0.5130 + {0.5000{\mathbb{i}}}} & {0.5366 + {0.2175{\mathbb{i}}}} \\ {{- 0.2395} + {0.0459{\mathbb{i}}}} & {0.4196 + {0.0255{\mathbb{i}}}} \end{matrix}$

${{ans}\left( {\text{:},\text{:},53} \right)} = \begin{matrix} {0.0848 + {0.2248{\mathbb{i}}}} & {0.5883 + {0.1906{\mathbb{i}}}} \\ {{- 0.2010} + {0.7357{\mathbb{i}}}} & {{- 0.0481} - {0.2343{\mathbb{i}}}} \\ {0.3862 + {0.2270{\mathbb{i}}}} & {0.2323 + {0.5375{\mathbb{i}}}} \\ {{- 0.3114} + {0.2509{\mathbb{i}}}} & {0.4661 + {0.0138{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},54} \right)} = \begin{matrix} {0.0848 + {0.2248{\mathbb{i}}}} & {{- 0.4550} - {0.1147{\mathbb{i}}}} \\ {{- 0.2010} + {0.7357{\mathbb{i}}}} & {{- 0.3923} + {0.0511{\mathbb{i}}}} \\ {0.3862 + {0.2270{\mathbb{i}}}} & {0.1654 - {0.1078{\mathbb{i}}}} \\ {{- 0.3114} + {0.2509{\mathbb{i}}}} & {0.6358 + {0.4245{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},55} \right)} = \begin{matrix} {0.0848 + {0.2248{\mathbb{i}}}} & {{- 0.5202} + {0.2628{\mathbb{i}}}} \\ {{- 0.2010} + {0.7357{\mathbb{i}}}} & {0.3107 - {0.3288{\mathbb{i}}}} \\ {0.3862 + {0.2270{\mathbb{i}}}} & {0.6089 + {0.2161{\mathbb{i}}}} \\ {{- 0.3114} + {0.2509{\mathbb{i}}}} & {{- 0.1325} - {0.1438{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},56} \right)} = \begin{matrix} {0.5883 + {0.1906{\mathbb{i}}}} & {{- 0.4550} - {0.1147{\mathbb{i}}}} \\ {{- 0.0481} - {0.2343{\mathbb{i}}}} & {{- 0.3923} + {0.0511{\mathbb{i}}}} \\ {0.2323 + {0.5375{\mathbb{i}}}} & {0.1654 - {0.1078{\mathbb{i}}}} \\ {0.4661 + {0.0138{\mathbb{i}}}} & {0.6358 + {0.4245{\mathbb{i}}}} \end{matrix}$

${{ans}\left( {\text{:},\text{:},57} \right)} = \begin{matrix} {0.5883 + {0.1906{\mathbb{i}}}} & {{- 0.5202} + {0.2628{\mathbb{i}}}} \\ {{- 0.0481} - {0.2343{\mathbb{i}}}} & {0.3107 - {0.3288{\mathbb{i}}}} \\ {0.2323 + {0.5375{\mathbb{i}}}} & {0.6089 + {0.2161{\mathbb{i}}}} \\ {0.4661 + {0.0138{\mathbb{i}}}} & {{- 0.1325} - {0.1438{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},58} \right)} = \begin{matrix} {{- 0.4550} - {0.1147{\mathbb{i}}}} & {{- 0.5202} + {0.2628{\mathbb{i}}}} \\ {{- 0.3923} + {0.0511{\mathbb{i}}}} & {0.3107 - {0.3288{\mathbb{i}}}} \\ {0.1654 - {0.1078{\mathbb{i}}}} & {0.6089 + {0.2161{\mathbb{i}}}} \\ {0.6358 + {0.4245{\mathbb{i}}}} & {{- 0.1325} - {0.1438{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},:,59} \right)} = \begin{matrix} {0.3968 - {0.0671{\mathbb{i}}}} & {{- 0.6008} - {0.3299{\mathbb{i}}}} \\ {{- 0.8289} - {0.0667{\mathbb{i}}}} & {{- 0.4011} - {0.1031{\mathbb{i}}}} \\ {0.1770 - {0.1687{\mathbb{i}}}} & {0.2785 + {0.3837{\mathbb{i}}}} \\ {{- 0.2814} + {0.0866{\mathbb{i}}}} & {0.3519 - {0.1000{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},60} \right)} = \begin{matrix} {0.3110 + {0.1356{\mathbb{i}}}} & {{- 0.2168} - {0.4540{\mathbb{i}}}} \\ {{- 0.2015} + {0.0416{\mathbb{i}}}} & {0.1102 - {0.2871{\mathbb{i}}}} \\ {{- 0.3325} - {0.1370{\mathbb{i}}}} & {0.0567 - {0.7634{\mathbb{i}}}} \\ {0.7347 - {0.4165{\mathbb{i}}}} & {{- 0.0390} - {0.2544{\mathbb{i}}}} \end{matrix}$

${{ans}\left( {\text{:},\text{:},61} \right)} = \begin{matrix} {0.0743 - {0.7846{\mathbb{i}}}} & {{- 0.2621} + {0.0798{\mathbb{i}}}} \\ {0.1307 - {0.3436{\mathbb{i}}}} & {0.4783 - {0.5634{\mathbb{i}}}} \\ {{- 0.3387} - {0.2385{\mathbb{i}}}} & {0.5212 + {0.1951{\mathbb{i}}}} \\ {{- 0.0962} + {0.2508{\mathbb{i}}}} & {{- 0.2418} + {0.1026{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},62} \right)} = \begin{matrix} {{- 0.4063} + {0.3394{\mathbb{i}}}} & {{- 0.2621} + {0.0798{\mathbb{i}}}} \\ {0.2466 + {0.0922{\mathbb{i}}}} & {0.4783 - {0.5634{\mathbb{i}}}} \\ {{- 0.4080} - {0.5055{\mathbb{i}}}} & {0.5212 + {0.1951{\mathbb{i}}}} \\ {{- 0.4767} - {0.0347{\mathbb{i}}}} & {{- 0.2418} + {0.1026{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},63} \right)} = \begin{matrix} {{- 0.4063} + {0.3394{\mathbb{i}}}} & {0.1421 + {0.0583{\mathbb{i}}}} \\ {0.2466 + {0.0922{\mathbb{i}}}} & {{- 0.0799} - {0.4928{\mathbb{i}}}} \\ {{- 0.4080} - {0.5055{\mathbb{i}}}} & {{- 0.2917} + {0.1078{\mathbb{i}}}} \\ {{- 0.4767} - {0.0347{\mathbb{i}}}} & {{- 0.0231} - {0.7937{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},64} \right)} = \begin{matrix} {{- 0.2621} + {0.0798{\mathbb{i}}}} & {0.1421 + {0.0583{\mathbb{i}}}} \\ {0.4783 - {0.5634{\mathbb{i}}}} & {{- 0.0799} - {0.4928{\mathbb{i}}}} \\ {0.5212 + {0.1951{\mathbb{i}}}} & {{- 0.2917} + {0.1078{\mathbb{i}}}} \\ {{- 0.2418} + {0.1026{\mathbb{i}}}} & {{- 0.0231} - {0.7937{\mathbb{i}}}} \end{matrix}$

(3) 6-Bit Rank 3 Codebook:

${{ans}\left( {\text{:},\text{:},1} \right)} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 \\ 0.5000 & {0.0000 + {0.5000{\mathbb{i}}}} & {{- 0.5000} + {0.0000{\mathbb{i}}}} \\ 0.5000 & {{- 0.5000} + {0.0000{\mathbb{i}}}} & {0.5000 - {0.0000{\mathbb{i}}}} \\ 0.5000 & {{- 0.0000} - {0.5000{\mathbb{i}}}} & {{- 0.5000} + {0.0000{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},2} \right)} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 \\ 0.5000 & {0.0000 + {0.5000{\mathbb{i}}}} & {{- 0.0000} - {0.5000{\mathbb{i}}}} \\ 0.5000 & {{- 0.5000} + {0.0000{\mathbb{i}}}} & {{- 0.5000} + {0.0000{\mathbb{i}}}} \\ 0.5000 & {{- 0.0000} - {0.5000{\mathbb{i}}}} & {0.0000 + {0.5000{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},3} \right)} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 \\ 0.5000 & {{- 0.5000} + {0.0000{\mathbb{i}}}} & {{- 0.0000} - {0.5000{\mathbb{i}}}} \\ 0.5000 & {0.5000 - {0.0000{\mathbb{i}}}} & {{- 0.5000} + {0.0000{\mathbb{i}}}} \\ 0.5000 & {{- 0.5000} + {0.0000{\mathbb{i}}}} & {0.0000 + {0.5000{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},4} \right)} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 \\ {0.0000 + {0.5000{\mathbb{i}}}} & {{- 0.5000} + {0.0000{\mathbb{i}}}} & {{- 0.0000} - {0.5000{\mathbb{i}}}} \\ {{- 0.5000} + {0.0000{\mathbb{i}}}} & {0.5000 - {0.0000{\mathbb{i}}}} & {{- 0.5000} + {0.0000{\mathbb{i}}}} \\ {{- 0.0000} - {0.5000{\mathbb{i}}}} & {{- 0.5000} + {0.0000{\mathbb{i}}}} & {0.0000 + {0.5000{\mathbb{i}}}} \end{matrix}$

${{ans}\left( {\text{:},\text{:},5} \right)} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 \\ {0.3536 + {0.3536{\mathbb{i}}}} & {{- 0.3536} + {0.3536{\mathbb{i}}}} & {{- 0.3536} - {0.3536{\mathbb{i}}}} \\ {0.0000 + {0.5000{\mathbb{i}}}} & {{- 0.0000} - {0.5000{\mathbb{i}}}} & {0.0000 + {0.5000{\mathbb{i}}}} \\ {{- 0.3536} + {0.3536{\mathbb{i}}}} & {0.3536 + {0.3536{\mathbb{i}}}} & {0.3536 - {0.3536`{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},6} \right)} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 \\ {0.3536 + {0.3536{\mathbb{i}}}} & {{- 0.3536} + {0.3536{\mathbb{i}}}} & {0.3536 - {0.3536{\mathbb{i}}}} \\ {0.0000 + {0.5000{\mathbb{i}}}} & {{- 0.0000} - {0.5000{\mathbb{i}}}} & {{- 0.0000} - {0.5000{\mathbb{i}}}} \\ {{- 0.3536} + {0.3536{\mathbb{i}}}} & {0.3536 + {0.3536{\mathbb{i}}}} & {{- 0.3536} - {0.3536`{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},7} \right)} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 \\ {0.3536 + {0.3536{\mathbb{i}}}} & {{- 0.3536} - {0.3536{\mathbb{i}}}} & {0.3536 - {0.3536{\mathbb{i}}}} \\ {0.0000 + {0.5000{\mathbb{i}}}} & {0.0000 + {0.5000{\mathbb{i}}}} & {{- 0.0000} - {0.5000{\mathbb{i}}}} \\ {{- 0.3536} + {0.3536{\mathbb{i}}}} & {0.3536 - {0.3536{\mathbb{i}}}} & {{- 0.3536} - {0.3536`{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},8} \right)} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 \\ {{- 0.3536} + {0.3536{\mathbb{i}}}} & {{- 0.3536} - {0.3536{\mathbb{i}}}} & {0.3536 - {0.3536{\mathbb{i}}}} \\ {{- 0.0000} - {0.5000{\mathbb{i}}}} & {0.0000 + {0.5000{\mathbb{i}}}} & {{- 0.0000} - {0.5000{\mathbb{i}}}} \\ {0.3536 + {0.3536{\mathbb{i}}}} & {0.3536 - {0.3536{\mathbb{i}}}} & {{- 0.3536} - {0.3536`{\mathbb{i}}}} \end{matrix}$

${{ans}\left( {\text{:},\text{:},9} \right)} = \begin{matrix} {0.4619 + {0.1913{\mathbb{i}}}} & {{- 0.1913} + {0.4619{\mathbb{i}}}} & {{- 0.4619} - {0.1913{\mathbb{i}}}} \\ {0.4455 - {0.2270{\mathbb{i}}}} & {0.4455 - {0.2270{\mathbb{i}}}} & {0.4455 - {0.2270{\mathbb{i}}}} \\ {{- 0.1167} + {0.4862{\mathbb{i}}}} & {0.4862 + {0.1167{\mathbb{i}}}} & {0.1167 - {0.4862{\mathbb{i}}}} \\ {0.2939 + {0.4045{\mathbb{i}}}} & {{- 0.2939} - {0.4045{\mathbb{i}}}} & {0.2939 + {0.4045{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},10} \right)} = \begin{matrix} {0.4619 + {0.1913{\mathbb{i}}}} & {{- 0.1913} + {0.4619{\mathbb{i}}}} & {0.1913 - {0.4619{\mathbb{i}}}} \\ {0.4455 - {0.2270{\mathbb{i}}}} & {0.4455 - {0.2270{\mathbb{i}}}} & {0.4455 - {0.2270{\mathbb{i}}}} \\ {{- 0.1167} + {0.4862{\mathbb{i}}}} & {0.4862 + {0.1167{\mathbb{i}}}} & {{- 0.4862} - {0.1167{\mathbb{i}}}} \\ {0.2939 + {0.4045{\mathbb{i}}}} & {{- 0.2939} - {0.4045{\mathbb{i}}}} & {{- 0.2939} - {0.4045{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},11} \right)} = \begin{matrix} {0.4619 + {0.1913{\mathbb{i}}}} & {{- 0.4619} - {0.1913{\mathbb{i}}}} & {0.1913 - {0.4619{\mathbb{i}}}} \\ {0.4455 - {0.2270{\mathbb{i}}}} & {0.4455 - {0.2270{\mathbb{i}}}} & {0.4455 - {0.2270{\mathbb{i}}}} \\ {{- 0.1167} + {0.4862{\mathbb{i}}}} & {0.1167 - {0.4862{\mathbb{i}}}} & {{- 0.4862} - {0.1167{\mathbb{i}}}} \\ {0.2939 + {0.4045{\mathbb{i}}}} & {0.2939 + {0.4045{\mathbb{i}}}} & {{- 0.2939} - {0.4045{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},12} \right)} = \begin{matrix} {{- 0.1913} + {0.4619{\mathbb{i}}}} & {{- 0.4619} - {0.1913{\mathbb{i}}}} & {0.1913 - {0.4619{\mathbb{i}}}} \\ {0.4455 - {0.2270{\mathbb{i}}}} & {0.4455 - {0.2270{\mathbb{i}}}} & {0.4455 - {0.2270{\mathbb{i}}}} \\ {0.4862 + {0.1167{\mathbb{i}}}} & {0.1167 - {0.4862{\mathbb{i}}}} & {{- 0.4862} - {0.1167{\mathbb{i}}}} \\ {{- 0.2939} - {0.4045{\mathbb{i}}}} & {0.2939 + {0.4045{\mathbb{i}}}} & {{- 0.2939} - {0.4045{\mathbb{i}}}} \end{matrix}$

${{ans}\left( {\text{:},\text{:},13} \right)} = \begin{matrix} {- 0.5000} & {- 0.5000} & {- 0.5000} \\ {0.4985 - {0.0392{\mathbb{i}}}} & {0.0392 + {0.4985{\mathbb{i}}}} & {{- 0.4985} + {0.0392{\mathbb{i}}}} \\ {0.4938 - {0.0782{\mathbb{i}}}} & {{- 0.4938} + {0.0782{\mathbb{i}}}} & {0.4938 - {0.0782{\mathbb{i}}}} \\ {{- 0.4862} + {0.1167{\mathbb{i}}}} & {0.1167 + {0.4862{\mathbb{i}}}} & {0.4862 - {0.1167{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},14} \right)} = \begin{matrix} {- 0.5000} & {- 0.5000} & {- 0.5000} \\ {0.4985 - {0.0392{\mathbb{i}}}} & {0.0392 + {0.4985{\mathbb{i}}}} & {{- 0.0392} - {0.4985{\mathbb{i}}}} \\ {0.4938 - {0.0782{\mathbb{i}}}} & {{- 0.4938} + {0.0782{\mathbb{i}}}} & {{- 0.4938} + {0.0782{\mathbb{i}}}} \\ {{- 0.4862} + {0.1167{\mathbb{i}}}} & {0.1167 + {0.4862{\mathbb{i}}}} & {{- 0.1167} - {0.4862{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},15} \right)} = \begin{matrix} {- 0.5000} & {- 0.5000} & {- 0.5000} \\ {0.4985 - {0.0392{\mathbb{i}}}} & {{- 0.4985} + {0.0392{\mathbb{i}}}} & {{- 0.0392} - {0.4985{\mathbb{i}}}} \\ {0.4938 - {0.0782{\mathbb{i}}}} & {0.4938 - {0.0782{\mathbb{i}}}} & {{- 0.4938} + {0.0782{\mathbb{i}}}} \\ {{- 0.4862} + {0.1167{\mathbb{i}}}} & {0.4862 - {0.1167{\mathbb{i}}}} & {{- 0.1167} - {0.4862{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},16} \right)} = \begin{matrix} {- 0.5000} & {- 0.5000} & {- 0.5000} \\ {0.0392 + {0.4985{\mathbb{i}}}} & {{- 0.4985} + {0.0392{\mathbb{i}}}} & {{- 0.0392} - {0.4985{\mathbb{i}}}} \\ {{- 0.4938} + {0.0782{\mathbb{i}}}} & {0.4938 - {0.0782{\mathbb{i}}}} & {{- 0.4938} + {0.0782{\mathbb{i}}}} \\ {0.1167 + {0.4862{\mathbb{i}}}} & {0.4862 - {0.1167{\mathbb{i}}}} & {{- 0.1167} - {0.4862{\mathbb{i}}}} \end{matrix}$

${{ans}\left( {\text{:},:,17} \right)} = \begin{matrix} 0.5000` & 0.5000 & 0.5000 \\ {0.4619 + {0.1913{\mathbb{i}}}} & {{- 0.1913} + {0.4619{\mathbb{i}}}} & {{- 0.4619} - {0.1913{\mathbb{i}}}} \\ {0.3536 + {0.3536{\mathbb{i}}}} & {{- 0.3536} - {0.3536{\mathbb{i}}}} & {0.3536 + {0.3536{\mathbb{i}}}} \\ {0.1913 + {0.4619{\mathbb{i}}}} & {0.4619 - {0.1913{\mathbb{i}}}} & {{- 0.1913} - {0.4619{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},18} \right)} = \begin{matrix} 0.5000` & 0.5000 & 0.5000 \\ {0.4619 + {0.1913{\mathbb{i}}}} & {{- 0.1913} + {0.4619{\mathbb{i}}}} & {0.1913 - {0.4619{\mathbb{i}}}} \\ {0.3536 + {0.3536{\mathbb{i}}}} & {{- 0.3536} - {0.3536{\mathbb{i}}}} & {{- 0.3536} - {0.3536{\mathbb{i}}}} \\ {0.1913 + {0.4619{\mathbb{i}}}} & {0.4619 - {0.1913{\mathbb{i}}}} & {{- 0.4619} + {0.1913{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},19} \right)} = \begin{matrix} 0.5000` & 0.5000 & 0.5000 \\ {0.4619 + {0.1913{\mathbb{i}}}} & {{- 0.4619} - {0.1913{\mathbb{i}}}} & {0.1913 - {0.4619{\mathbb{i}}}} \\ {0.3536 + {0.3536{\mathbb{i}}}} & {0.3536 + {0.3536{\mathbb{i}}}} & {{- 0.3536} - {0.3536{\mathbb{i}}}} \\ {0.1913 + {0.4619{\mathbb{i}}}} & {{- 0.1913} - {0.4619{\mathbb{i}}}} & {{- 0.4619} + {0.1913{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},20} \right)} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 \\ {{- 0.1913} + {0.4619{\mathbb{i}}}} & {{- 0.4619} - {0.1913{\mathbb{i}}}} & {0.1913 - {0.4619{\mathbb{i}}}} \\ {{- 0.3536} - {0.3536{\mathbb{i}}}} & {0.3536 + {0.3536{\mathbb{i}}}} & {{- 0.3536} - {0.3536{\mathbb{i}}}} \\ {0.4619 - {0.1913{\mathbb{i}}}} & {{- 0.1913} - {0.4619{\mathbb{i}}}} & {{- 0.4619} + {0.1913{\mathbb{i}}}} \end{matrix}$

${{ans}\left( {\text{:},\text{:},21} \right)} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 \\ {0.1913 + {0.4619{\mathbb{i}}}} & {{- 0.4619} + {0.1913{\mathbb{i}}}} & {{- 0.1913} - {0.4619{\mathbb{i}}}} \\ {{- 0.3536} + {0.3536{\mathbb{i}}}} & {0.3536 - {0.3536{\mathbb{i}}}} & {{- 0.3536} + {0.3536{\mathbb{i}}}} \\ {{- 0.4619} - {0.1913{\mathbb{i}}}} & {{- 0.1913} + {0.4619{\mathbb{i}}}} & {0.4619 + {0.1913{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},22} \right)} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 \\ {0.1913 + {0.4619{\mathbb{i}}}} & {{- 0.4619} + {0.1913{\mathbb{i}}}} & {0.4619 - {0.1913{\mathbb{i}}}} \\ {{- 0.3536} + {0.3536{\mathbb{i}}}} & {0.3536 - {0.3536{\mathbb{i}}}} & {0.3536 - {0.3536{\mathbb{i}}}} \\ {{- 0.4619} - {0.1913{\mathbb{i}}}} & {{- 0.1913} + {0.4619{\mathbb{i}}}} & {0.1913 - {0.4619{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},23} \right)} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 \\ {0.1913 + {0.4619{\mathbb{i}}}} & {{- 0.1913} - {0.4619{\mathbb{i}}}} & {0.4619 - {0.1913{\mathbb{i}}}} \\ {{- 0.3536} + {0.3536{\mathbb{i}}}} & {{- 0.3536} + {0.3536{\mathbb{i}}}} & {0.3536 - {0.3536{\mathbb{i}}}} \\ {{- 0.4619} - {0.1913{\mathbb{i}}}} & {0.4619 + {0.1913{\mathbb{i}}}} & {0.1913 - {0.4619{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},24} \right)} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 \\ {{- 0.4619} + {0.1913{\mathbb{i}}}} & {{- 0.1913} - {0.4619{\mathbb{i}}}} & {0.4619 - {0.1913{\mathbb{i}}}} \\ {0.3536 - {0.3536{\mathbb{i}}}} & {{- 0.3536} + {0.3536{\mathbb{i}}}} & {0.3536 - {0.3536{\mathbb{i}}}} \\ {{- 0.1913} + {0.4619{\mathbb{i}}}} & {0.4619 + {0.1913{\mathbb{i}}}} & {0.1913 - {0.4619{\mathbb{i}}}} \end{matrix}$

${{ans}\left( {\text{:},\text{:},25} \right)} = \begin{matrix} {0.2437 + {0.4837{\mathbb{i}}}} & {0.4258 + {0.0076{\mathbb{i}}}} & {0.5210 - {0.4265{\mathbb{i}}}} \\ {0.5740 - {0.3102{\mathbb{i}}}} & {0.0596 + {0.3222{\mathbb{i}}}} & {0.2922 + {0.0454{\mathbb{i}}}} \\ {0.2823 - {0.3783{\mathbb{i}}}} & {{- 0.4186} + {0.1136{\mathbb{i}}}} & {0.3628 + {0.1522{\mathbb{i}}}} \\ {0.2062 + {0.1248{\mathbb{i}}}} & {0.1513 + {0.7073{\mathbb{i}}}} & {{- 0.5498} - {0.0464{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},26} \right)} = \begin{matrix} {0.2437 + {0.4837{\mathbb{i}}}} & {0.4258 + {0.0076{\mathbb{i}}}} & {{- 0.2364} - {0.1268{\mathbb{i}}}} \\ {0.5740 - {0.3102{\mathbb{i}}}} & {0.0596 + {0.3222{\mathbb{i}}}} & {0.6147 + {0.0409{\mathbb{i}}}} \\ {0.2823 - {0.3783{\mathbb{i}}}} & {{- 0.4186} + {0.1136{\mathbb{i}}}} & {{- 0.6500} - {0.1085{\mathbb{i}}}} \\ {0.2062 + {0.1248{\mathbb{i}}}} & {0.1513 + {0.7073{\mathbb{i}}}} & {{- 0.2340} - {0.2440{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},27} \right)} = \begin{matrix} {0.2437 + {0.4837{\mathbb{i}}}} & {0.5210 - {0.4265{\mathbb{i}}}} & {{- 0.2364} - {0.1268{\mathbb{i}}}} \\ {0.5740 - {0.3102{\mathbb{i}}}} & {0.2922 + {0.0454{\mathbb{i}}}} & {0.6147 + {0.0409{\mathbb{i}}}} \\ {0.2823 - {0.3783{\mathbb{i}}}} & {0.3628 + {0.1522{\mathbb{i}}}} & {{- 0.6500} - {0.1085{\mathbb{i}}}} \\ {0.2062 + {0.1248{\mathbb{i}}}} & {{- 0.5498} - {0.0464{\mathbb{i}}}} & {{- 0.2340} - {0.2440{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},28} \right)} = \begin{matrix} {0.4258 + {0.0076{\mathbb{i}}}} & {0.5210 - {0.4265{\mathbb{i}}}} & {{- 0.2364} - {0.1268{\mathbb{i}}}} \\ {0.0596 + {0.3222{\mathbb{i}}}} & {0.2922 + {0.0454{\mathbb{i}}}} & {0.6147 + {0.0409{\mathbb{i}}}} \\ {{- 0.4186} + {0.1136{\mathbb{i}}}} & {0.3628 + {0.1522{\mathbb{i}}}} & {{- 0.6500} - {0.1085{\mathbb{i}}}} \\ {0.1513 + {0.7073{\mathbb{i}}}} & {{- 0.5498} - {0.0464{\mathbb{i}}}} & {{- 0.2340} - {0.2440{\mathbb{i}}}} \end{matrix}$

${{ans}\left( {\text{:},\text{:},29} \right)} = \begin{matrix} {0.0591 - {0.4953{\mathbb{i}}}} & {0.0742 + {0.2714{\mathbb{i}}}} & {0.3370 + {0.2000{\mathbb{i}}}} \\ {0.5545 + {0.3908{\mathbb{i}}}} & {{- 0.0591} - {0.0963{\mathbb{i}}}} & {0.2158 + {0.4089{\mathbb{i}}}} \\ {0.2593 - {0.0234{\mathbb{i}}}} & {{- 0.5800} - {0.1645{\mathbb{i}}}} & {{- 0.2363} - {0.5965{\mathbb{i}}}} \\ {0.3300 + {0.3380{\mathbb{i}}}} & {0.4619 + {0.5755{\mathbb{i}}}} & {0.0175 - {0.4698{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},30} \right)} = \begin{matrix} {0.0591 - {0.4953{\mathbb{i}}}} & {0.0742 + {0.2714{\mathbb{i}}}} & {{- 0.1041} - {0.7125{\mathbb{i}}}} \\ {0.5545 + {0.3908{\mathbb{i}}}} & {{- 0.0591} - {0.0963{\mathbb{i}}}} & {{- 0.5589} + {0.0299{\mathbb{i}}}} \\ {0.2593 - {0.0234{\mathbb{i}}}} & {{- 0.5800} - {0.1645{\mathbb{i}}}} & {{- 0.2149} - {0.3331{\mathbb{i}}}} \\ {0.3300 + {0.3380{\mathbb{i}}}} & {0.4619 + {0.5755{\mathbb{i}}}} & {0.0748 - {0.0748{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},31} \right)} = \begin{matrix} {0.0591 - {0.4953{\mathbb{i}}}} & {0.3370 + {0.2000{\mathbb{i}}}} & {{- 0.1041} - {0.7125{\mathbb{i}}}} \\ {0.5545 + {0.3908{\mathbb{i}}}} & {0.2158 + {0.4089{\mathbb{i}}}} & {{- 0.5589} + {0.0299{\mathbb{i}}}} \\ {0.2593 - {0.0234{\mathbb{i}}}} & {{- 0.2363} - {0.5965{\mathbb{i}}}} & {{- 0.2149} - {0.3331{\mathbb{i}}}} \\ {0.3300 + {0.3380{\mathbb{i}}}} & {0.0175 - {0.4698{\mathbb{i}}}} & {0.0748 - {0.0748{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},32} \right)} = \begin{matrix} {0.0742 + {0.2714{\mathbb{i}}}} & {0.3370 + {0.2000{\mathbb{i}}}} & {{- 0.1041} - {0.7125{\mathbb{i}}}} \\ {{- 0.0591} - {0.0963{\mathbb{i}}}} & {0.2158 + {0.4089{\mathbb{i}}}} & {{- 0.5589} + {0.0299{\mathbb{i}}}} \\ {{- 0.5800} - {0.1645{\mathbb{i}}}} & {{- 0.2363} - {0.5965{\mathbb{i}}}} & {{- 0.2149} - {0.3331{\mathbb{i}}}} \\ {0.4619 + {0.5755{\mathbb{i}}}} & {0.0175 - {0.4698{\mathbb{i}}}} & {0.0748 - {0.0748{\mathbb{i}}}} \end{matrix}$

${{ans}\left( {\text{:},\text{:},33} \right)} = \begin{matrix} {0.5019 - {0.2171{\mathbb{i}}}} & {0.2121 - {0.4391{\mathbb{i}}}} & {0.4649 + {0.1545{\mathbb{i}}}} \\ {0.0570 + {0.3132{\mathbb{i}}}} & {0.3637 + {0.1907{\mathbb{i}}}} & {0.5971 - {0.5122{\mathbb{i}}}} \\ {0.2569 - {0.1797{\mathbb{i}}}} & {0.0539 - {0.3157{\mathbb{i}}}} & {{- 0.0025} + {0.3671{\mathbb{i}}}} \\ {0.6949 + {0.1358{\mathbb{i}}}} & {{- 0.5835} + {0.3879{\mathbb{i}}}} & {0.0058 - {0.0797{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},34} \right)} = \begin{matrix} {0.5019 - {0.2171{\mathbb{i}}}} & {0.2121 - {0.4391{\mathbb{i}}}} & {{- 0.4709} + {0.0383{\mathbb{i}}}} \\ {0.0570 + {0.3132{\mathbb{i}}}} & {0.3637 + {0.1907{\mathbb{i}}}} & {0.3285 + {0.0569{\mathbb{i}}}} \\ {0.2569 - {0.1797{\mathbb{i}}}} & {0.0539 - {0.3157{\mathbb{i}}}} & {0.8042 + {0.1327{\mathbb{i}}}} \\ {0.6949 + {0.1358{\mathbb{i}}}} & {{- 0.5835} + {0.3879{\mathbb{i}}}} & {0.0341 + {0.0124{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},35} \right)} = \begin{matrix} {0.5019 - {0.2171{\mathbb{i}}}} & {0.4649 + {0.1545{\mathbb{i}}}} & {{- 0.4709} + {0.0383{\mathbb{i}}}} \\ {0.0570 + {0.3132{\mathbb{i}}}} & {0.5971 - {0.5122{\mathbb{i}}}} & {0.3285 + {0.0569{\mathbb{i}}}} \\ {0.2569 - {0.1797{\mathbb{i}}}} & {{- 0.0025} + {0.3671{\mathbb{i}}}} & {0.8042 + {0.1327{\mathbb{i}}}} \\ {0.6949 + {0.1358{\mathbb{i}}}} & {0.0058 - {0.0797{\mathbb{i}}}} & {0.0341 + {0.0124{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},36} \right)} = \begin{matrix} {0.2121 - {0.4391{\mathbb{i}}}} & {0.4649 + {0.1545{\mathbb{i}}}} & {{- 0.4709} + {0.0383{\mathbb{i}}}} \\ {0.3637 + {0.1907{\mathbb{i}}}} & {0.5971 - {0.5122{\mathbb{i}}}} & {0.3285 + {0.0569{\mathbb{i}}}} \\ {0.0539 - {0.3157{\mathbb{i}}}} & {{- 0.0025} + {0.3671{\mathbb{i}}}} & {0.8042 + {0.1327{\mathbb{i}}}} \\ {{- 0.5835} + {0.3879{\mathbb{i}}}} & {0.0058 - {0.0797{\mathbb{i}}}} & {0.0341 + {0.0124{\mathbb{i}}}} \end{matrix}$

${{ans}\left( {\text{:},\text{:},37} \right)} = \begin{matrix} {0.2199 + {0.3199{\mathbb{i}}}} & {0.6305 - {0.4345{\mathbb{i}}}} & {{- 0.0525} + {0.2357{\mathbb{i}}}} \\ {0.5983 + {0.1264{\mathbb{i}}}} & {{- 0.0683} + {0.2985{\mathbb{i}}}} & {0.5833 + {0.2614{\mathbb{i}}}} \\ {{- 0.5576} - {0.2903{\mathbb{i}}}} & {{- 0.0495} - {0.3235{\mathbb{i}}}} & {0.3552 + {0.3750{\mathbb{i}}}} \\ {{- 0.0940} - {0.2672{\mathbb{i}}}} & {{- 0.0481} + {0.4588{\mathbb{i}}}} & {0.0118 + {0.5160{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},38} \right)} = \begin{matrix} {0.2199 + {0.3199{\mathbb{i}}}} & {0.6305 - {0.4345{\mathbb{i}}}} & {{- 0.3110} - {0.3286{\mathbb{i}}}} \\ {0.5983 + {0.1264{\mathbb{i}}}} & {{- 0.0683} + {0.2985{\mathbb{i}}}} & {0.2969 - {0.1888{\mathbb{i}}}} \\ {{- 0.5576} - {0.2903{\mathbb{i}}}} & {{- 0.0495} - {0.3235{\mathbb{i}}}} & {0.2888 - {0.3841{\mathbb{i}}}} \\ {{- 0.0940} - {0.2672{\mathbb{i}}}} & {{- 0.0481} + {0.4588{\mathbb{i}}}} & {{- 0.6633} - {0.0260{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},39} \right)} = \begin{matrix} {0.2199 + {0.3199{\mathbb{i}}}} & {{- 0.0525} + {0.2357{\mathbb{i}}}} & {{- 0.3110} - {0.3286{\mathbb{i}}}} \\ {0.5983 + {0.1264{\mathbb{i}}}} & {0.5833 + {0.2614{\mathbb{i}}}} & {0.2969 - {0.1888{\mathbb{i}}}} \\ {{- 0.5576} - {0.2903{\mathbb{i}}}} & {0.3552 + {0.3750{\mathbb{i}}}} & {0.2888 - {0.3841{\mathbb{i}}}} \\ {{- 0.0940} - {0.2672{\mathbb{i}}}} & {0.0118 + {0.5160{\mathbb{i}}}} & {{- 0.6633} - {0.0260{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},40} \right)} = \begin{matrix} {0.6305 - {0.4345{\mathbb{i}}}} & {{- 0.0525} + {0.2357{\mathbb{i}}}} & {{- 0.3110} - {0.3286{\mathbb{i}}}} \\ {{- 0.0683} + {0.2985{\mathbb{i}}}} & {0.5833 + {0.2614{\mathbb{i}}}} & {0.2969 - {0.1888{\mathbb{i}}}} \\ {{- 0.0495} - {0.3235{\mathbb{i}}}} & {0.3552 + {0.3750{\mathbb{i}}}} & {0.2888 - {0.3841{\mathbb{i}}}} \\ {{- 0.0481} + {0.4588{\mathbb{i}}}} & {0.0118 + {0.5160{\mathbb{i}}}} & {{- 0.6633} - {0.0260{\mathbb{i}}}} \end{matrix}$

${{ans}\left( {\text{:},\text{:},41} \right)} = \begin{matrix} {0.4534 + {0.1075{\mathbb{i}}}} & {0.2632 + {0.4348{\mathbb{i}}}} & {0.1838 + {0.6429{\mathbb{i}}}} \\ {{- 0.0163} + {0.7556{\mathbb{i}}}} & {0.3698 - {0.0222{\mathbb{i}}}} & {0.0055 - {0.3112{\mathbb{i}}}} \\ {0.0681 + {0.3202{\mathbb{i}}}} & {{- 0.4066} - {0.6110{\mathbb{i}}}} & {0.3882 + {0.2198{\mathbb{i}}}} \\ {0.1259 + {0.2977{\mathbb{i}}}} & {0.0347 + {0.2542{\mathbb{i}}}} & {0.4482 - {0.2369{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},42} \right)} = \begin{matrix} {0.4534 + {0.1075{\mathbb{i}}}} & {0.2632 + {0.4348{\mathbb{i}}}} & {0.2307 + {0.1558{\mathbb{i}}}} \\ {{- 0.0163} + {0.7556{\mathbb{i}}}} & {0.3698 - {0.0222{\mathbb{i}}}} & {0.3467 - {0.2728{\mathbb{i}}}} \\ {0.0681 + {0.3202{\mathbb{i}}}} & {{- 0.4066} - {0.6110{\mathbb{i}}}} & {0.1705 + {0.3551{\mathbb{i}}}} \\ {0.1259 + {0.2977{\mathbb{i}}}} & {0.0347 + {0.2542{\mathbb{i}}}} & {{- 0.7319} + {0.1924{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},43} \right)} = \begin{matrix} {0.4534 + {0.1075{\mathbb{i}}}} & {0.1838 + {0.6429{\mathbb{i}}}} & {0.2307 + {0.1558{\mathbb{i}}}} \\ {{- 0.0163} + {0.7556{\mathbb{i}}}} & {0.0055 - {0.3112{\mathbb{i}}}} & {0.3467 - {0.2728{\mathbb{i}}}} \\ {0.0681 + {0.3202{\mathbb{i}}}} & {0.3882 + {0.2198{\mathbb{i}}}} & {0.1705 + {0.3551{\mathbb{i}}}} \\ {0.1259 + {0.2977{\mathbb{i}}}} & {0.4482 - {0.2369{\mathbb{i}}}} & {{- 0.7319} + {0.1924{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},44} \right)} = \begin{matrix} {0.2632 + {0.4348{\mathbb{i}}}} & {0.1838 + {0.6429{\mathbb{i}}}} & {0.2307 + {0.1558{\mathbb{i}}}} \\ {0.3698 - {0.0222{\mathbb{i}}}} & {0.0055 - {0.3112{\mathbb{i}}}} & {0.3467 - {0.2728{\mathbb{i}}}} \\ {{- 0.4066} - {0.6110{\mathbb{i}}}} & {0.3882 + {0.2198{\mathbb{i}}}} & {0.1705 + {0.3551{\mathbb{i}}}} \\ {0.0347 + {0.2542{\mathbb{i}}}} & {0.4482 - {0.2369{\mathbb{i}}}} & {{- 0.7319} + {0.1924{\mathbb{i}}}} \end{matrix}$

${{ans}\left( {\text{:},\text{:},45} \right)} = \begin{matrix} {0.0534 - {0.5633{\mathbb{i}}}} & {{- 0.4493} + {0.3172{\mathbb{i}}}} & {{- 0.3346} + {0.1479`{\mathbb{i}}}} \\ {{- 0.1472} - {0.1184{\mathbb{i}}}} & {{- 0.5422} + {0.3203{\mathbb{i}}}} & {0.2302 - {0.4770{\mathbb{i}}}} \\ {{- 0.7445} + {0.0533{\mathbb{i}}}} & {{- 0.1168} - {0.4073{\mathbb{i}}}} & {{- 0.1288} - {0.3642{\mathbb{i}}}} \\ {0.2881 - {0.0638{\mathbb{i}}}} & {0.3438 + {0.0562{\mathbb{i}}}} & {{- 0.1358} - {0.6465{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},46} \right)} = \begin{matrix} {0.0534 - {0.5633{\mathbb{i}}}} & {{- 0.4493} + {0.3172{\mathbb{i}}}} & {{- 0.4868} - {0.0811{\mathbb{i}}}} \\ {{- 0.1472} - {0.1184{\mathbb{i}}}} & {{- 0.5422} + {0.3203{\mathbb{i}}}} & {0.4913 + {0.2140{\mathbb{i}}}} \\ {{- 0.7445} + {0.0533{\mathbb{i}}}} & {{- 0.1168} - {0.4073{\mathbb{i}}}} & {{- 0.2957} - {0.1634{\mathbb{i}}}} \\ {0.2881 - {0.0638{\mathbb{i}}}} & {0.3438 + {0.0562{\mathbb{i}}}} & {{- 0.3579} + {0.4766{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},47} \right)} = \begin{matrix} {0.0534 - {0.5633{\mathbb{i}}}} & {{- 0.3346} + {0.1479{\mathbb{i}}}} & {{- 0.4868} - {0.0811{\mathbb{i}}}} \\ {{- 0.1472} - {0.1184{\mathbb{i}}}} & {0.2302 - {0.4770{\mathbb{i}}}} & {0.4913 + {0.2140{\mathbb{i}}}} \\ {{- 0.7445} + {0.0533{\mathbb{i}}}} & {{- 0.1288} - {0.3642{\mathbb{i}}}} & {{- 0.2957} - {0.1634{\mathbb{i}}}} \\ {0.2881 - {0.0638{\mathbb{i}}}} & {{- 0.1358} - {0.6465{\mathbb{i}}}} & {{- 0.3579} + {0.4766{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},48} \right)} = \begin{matrix} {{- 0.4493} + {0.3172{\mathbb{i}}}} & {{- 0.3346} + {0.1479{\mathbb{i}}}} & {{- 0.4868} - {0.0811{\mathbb{i}}}} \\ {{- 0.5422} + {0.3203{\mathbb{i}}}} & {0.2302 - {0.4770{\mathbb{i}}}} & {0.4913 + {0.2140{\mathbb{i}}}} \\ {{- 0.1168} - {0.4073{\mathbb{i}}}} & {{- 0.1288} - {0.3642`{\mathbb{i}}}} & {{- 0.2957} - {0.1634{\mathbb{i}}}} \\ {0.3438 + {0.0562{\mathbb{i}}}} & {{- 0.1358} - {0.6465{\mathbb{i}}}} & {{- 0.3579} + {0.4766{\mathbb{i}}}} \end{matrix}$

${{ans}\left( {\text{:},\text{:},49} \right)} = \begin{matrix} {0.4192 + {0.3317{\mathbb{i}}}} & {0.6462 - {0.3049{\mathbb{i}}}} & {0.1765 - {0.0924{\mathbb{i}}}} \\ {{- 0.2340} + {0.0529{\mathbb{i}}}} & {{- 0.0388} - {0.4789{\mathbb{i}}}} & {0.6212 - {0.0425{\mathbb{i}}}} \\ {0.0491 - {0.2199{\mathbb{i}}}} & {{- 0.2962} + {0.1142{\mathbb{i}}}} & {0.5130 + {0.5000{\mathbb{i}}}} \\ {{- 0.7746} + {0.0772{\mathbb{i}}}} & {0.1089 - {0.3821{\mathbb{i}}}} & {{- 0.2395} + {0.0459{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},50} \right)} = \begin{matrix} {0.4192 + {0.3317{\mathbb{i}}}} & {0.6462 - {0.3049{\mathbb{i}}}} & {0.3680 + {0.1691{\mathbb{i}}}} \\ {{- 0.2340} + {0.0529{\mathbb{i}}}} & {{- 0.0388} - {0.4789{\mathbb{i}}}} & {{- 0.5223} + {0.2262{\mathbb{i}}}} \\ {0.0491 - {0.2199{\mathbb{i}}}} & {{- 0.2962} + {0.1142{\mathbb{i}}}} & {0.5366 + {0.2175{\mathbb{i}}}} \\ {{- 0.7746} + {0.0772{\mathbb{i}}}} & {0.1089 - {0.3821{\mathbb{i}}}} & {0.4196 + {0.0255{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},51} \right)} = \begin{matrix} {0.4192 + {0.3317{\mathbb{i}}}} & {0.1765 - {0.0924{\mathbb{i}}}} & {0.3680 + {0.1691{\mathbb{i}}}} \\ {{- 0.2340} + {0.0529{\mathbb{i}}}} & {0.6212 - {0.0425{\mathbb{i}}}} & {{- 0.5223} + {0.2262{\mathbb{i}}}} \\ {0.0491 - {0.2199{\mathbb{i}}}} & {0.5130 + {0.5000{\mathbb{i}}}} & {0.5366 + {0.2175{\mathbb{i}}}} \\ {{- 0.7746} + {0.0772{\mathbb{i}}}} & {{- 0.2395} + {0.0459{\mathbb{i}}}} & {0.4196 + {0.0255{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},52} \right)} = \begin{matrix} {0.6462 - {0.3049{\mathbb{i}}}} & {0.1765 - {0.0924{\mathbb{i}}}} & {0.3680 + {0.1691{\mathbb{i}}}} \\ {{- 0.0388} - {0.4789{\mathbb{i}}}} & {0.6212 - {0.0425{\mathbb{i}}}} & {{- 0.5223} + {0.2262{\mathbb{i}}}} \\ {{- 0.2962} + {0.1142{\mathbb{i}}}} & {0.5130 + {0.5000{\mathbb{i}}}} & {0.5366 + {0.2175{\mathbb{i}}}} \\ {0.1089 - {0.3821{\mathbb{i}}}} & {{- 0.2395} + {0.0459{\mathbb{i}}}} & {0.4196 + {0.0255{\mathbb{i}}}} \end{matrix}$

${{ans}\left( {\text{:},\text{:},53} \right)} = \begin{matrix} {0.0848 + {0.2248{\mathbb{i}}}} & {0.5883 + {0.1906{\mathbb{i}}}} & {{- 0.4550} - {0.1147{\mathbb{i}}}} \\ {{- 0.2010} + {0.7357{\mathbb{i}}}} & {{- 0.0481} - {0.2343`{\mathbb{i}}}} & {{- 0.3923} + {0.0511{\mathbb{i}}}} \\ {0.3862 + {0.2270{\mathbb{i}}}} & {0.2323 + {0.5375{\mathbb{i}}}} & {0.1654 - {0.1078{\mathbb{i}}}} \\ {{- 0.3114} + {0.2509{\mathbb{i}}}} & {0.4661 + {0.0138{\mathbb{i}}}} & {0.6358 + {0.4245{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},54} \right)} = \begin{matrix} {0.0848 + {0.2248{\mathbb{i}}}} & {0.5883 + {0.1906{\mathbb{i}}}} & {{- 0.5202} + {0.2628{\mathbb{i}}}} \\ {{- 0.2010} + {0.7357{\mathbb{i}}}} & {{- 0.0481} - {0.2343{\mathbb{i}}}} & {0.3107 - {0.3288{\mathbb{i}}}} \\ {0.3862 + {0.2270{\mathbb{i}}}} & {0.2323 + {0.5375{\mathbb{i}}}} & {0.6089 + {0.2161{\mathbb{i}}}} \\ {{- 0.3114} + {0.2509{\mathbb{i}}}} & {0.4661 + {0.0138{\mathbb{i}}}} & {{- 0.1325} - {0.1438{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},55} \right)} = \begin{matrix} {0.0848 + {0.2248{\mathbb{i}}}} & {{- 0.4550} - {0.1147{\mathbb{i}}}} & {{- 0.5202} + {0.2628{\mathbb{i}}}} \\ {{- 0.2010} + {0.7357{\mathbb{i}}}} & {{- 0.3923} + {0.0511{\mathbb{i}}}} & {0.3107 - {0.3288{\mathbb{i}}}} \\ {0.3862 + {0.2270`{\mathbb{i}}}} & {0.1654 - {0.1078{\mathbb{i}}}} & {0.6089 + {0.2161{\mathbb{i}}}} \\ {{- 0.3114} + {0.2509{\mathbb{i}}}} & {0.6358 + {0.4245{\mathbb{i}}}} & {{- 0.1325} - {0.1438{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},56} \right)} = \begin{matrix} {0.5883 + {0.1906{\mathbb{i}}}} & {{- 0.4550} - {0.1147{\mathbb{i}}}} & {{- 0.5202} + {0.2628{\mathbb{i}}}} \\ {{- 0.0481} - {0.2343{\mathbb{i}}}} & {{- 0.3923} + {0.0511{\mathbb{i}}}} & {0.3107 - {0.3288{\mathbb{i}}}} \\ {0.2323 + {0.5375{\mathbb{i}}}} & {0.1654 - {0.1078{\mathbb{i}}}} & {0.6089 + {0.2161{\mathbb{i}}}} \\ {0.4661 + {0.0138{\mathbb{i}}}} & {0.6358 + {0.4245{\mathbb{i}}}} & {{- 0.1325} - {0.1438{\mathbb{i}}}} \end{matrix}$

$\;{{{ans}\left( {\text{:},\text{:},57} \right)} = \begin{matrix} {0.3968 - {0.0671{\mathbb{i}}}} & {0.3110 + {0.1356{\mathbb{i}}}} & {{- 0.2168} - {0.4540{\mathbb{i}}}} \\ {{- 0.8289} - {0.0667{\mathbb{i}}}} & {{- 0.2015} + {0.0416{\mathbb{i}}}} & {0.1102 - {0.2871{\mathbb{i}}}} \\ {0.1770 - {0.1687{\mathbb{i}}}} & {{- 0.3325} - {0.1370{\mathbb{i}}}} & {0.0567 - {0.7634{\mathbb{i}}}} \\ {{- 0.2814} + {0.0866{\mathbb{i}}}} & {0.7347 - {0.4165{\mathbb{i}}}} & {{- 0.0390} - {0.2544{\mathbb{i}}}} \end{matrix}}$ ${{ans}\left( {\text{:},\text{:},58} \right)} = \begin{matrix} {0.3968 - {0.0671`{\mathbb{i}}}} & {0.3110 + {0.1356{\mathbb{i}}}} & {{- 0.6008} - {0.3299{\mathbb{i}}}} \\ {{- 0.8289} - {0.0667{\mathbb{i}}}} & {{- 0.2015} + {0.0416{\mathbb{i}}}} & {{- 0.4011} - {0.1031{\mathbb{i}}}} \\ {0.1770 - {0.1687{\mathbb{i}}}} & {{- 0.3325} - {0.1370{\mathbb{i}}}} & {0.2785 + {0.3837{\mathbb{i}}}} \\ {{- 0.2814} + {0.0866{\mathbb{i}}}} & {0.7347 - {0.4165{\mathbb{i}}}} & {0.3519 - {0.1000{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},59} \right)} = \begin{matrix} {0.3968 - {0.0671{\mathbb{i}}}} & {{- 0.2168} - {0.4540{\mathbb{i}}}} & {{- 0.6008} - {0.3299{\mathbb{i}}}} \\ {{- 0.8289} - {0.0667{\mathbb{i}}}} & {0.1102 - {0.2871{\mathbb{i}}}} & {{- 0.4011} - {0.1031{\mathbb{i}}}} \\ {0.1770 - {0.1687{\mathbb{i}}}} & {0.0567 - {0.7634{\mathbb{i}}}} & {0.2785 + {0.3837`{\mathbb{i}}}} \\ {{- 0.2814} + {0.0866{\mathbb{i}}}} & {{- 0.0390} - {0.2544{\mathbb{i}}}} & {0.3519 - {0.1000{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},60} \right)} = \begin{matrix} {0.3110 + {0.1356{\mathbb{i}}}} & {{- 0.2168} - {0.4540{\mathbb{i}}}} & {{- 0.6008} - {0.3299{\mathbb{i}}}} \\ {{- 0.2015} + {0.0416{\mathbb{i}}}} & {0.1102 - {0.2871{\mathbb{i}}}} & {{- 0.4011} - {0.1031{\mathbb{i}}}} \\ {{- 0.3325} - {0.1370{\mathbb{i}}}} & {0.0567 - {0.7634{\mathbb{i}}}} & {0.2785 + {0.3837{\mathbb{i}}}} \\ {0.7347 - {0.4165{\mathbb{i}}}} & {{- 0.0390} - {0.2544{\mathbb{i}}}} & {0.3519 - {0.1000{\mathbb{i}}}} \end{matrix}$

$\;{{{ans}\left( {\text{:},\text{:},61} \right)} = \begin{matrix} {0.0743 - {0.7846{\mathbb{i}}}} & {{- 0.4063} + {0.3394{\mathbb{i}}}} & {{- 0.2621} + {0.0798{\mathbb{i}}}} \\ {0.1307 - {0.3436{\mathbb{i}}}} & {0.2466 + {0.0922{\mathbb{i}}}} & {0.4783 - {0.5634{\mathbb{i}}}} \\ {{- 0.3387} - {0.2385{\mathbb{i}}}} & {{- 0.4080} - {0.5055{\mathbb{i}}}} & {0.5212 + {0.1951{\mathbb{i}}}} \\ {{- 0.0962} + {0.2508{\mathbb{i}}}} & {{- 0.4767} - {0.0347{\mathbb{i}}}} & {{- 0.2418} + {0.1026{\mathbb{i}}}} \end{matrix}}$ ${{ans}\left( {\text{:},\text{:},62} \right)} = \begin{matrix} {0.0743 - {0.7846{\mathbb{i}}}} & {{- 0.4063} + {0.3394{\mathbb{i}}}} & {0.1421 + {0.0583{\mathbb{i}}}} \\ {0.1307 - {0.3436{\mathbb{i}}}} & {0.2466 + {0.0922{\mathbb{i}}}} & {{- 0.0799} - {0.4928{\mathbb{i}}}} \\ {{- 0.3387} - {0.2385{\mathbb{i}}}} & {{- 0.4080} - {0.5055{\mathbb{i}}}} & {{- 0.2917} + {0.1078{\mathbb{i}}}} \\ {{- 0.0962} + {0.2508{\mathbb{i}}}} & {{- 0.4767} - {0.0347{\mathbb{i}}}} & {{- 0.0231} - {0.7937{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},63} \right)} = \begin{matrix} {0.0743 - {0.7846{\mathbb{i}}}} & {{- 0.2621} + {0.0798{\mathbb{i}}}} & {0.1421 + {0.0583{\mathbb{i}}}} \\ {0.1307 - {0.3436{\mathbb{i}}}} & {0.4783 - {0.5634{\mathbb{i}}}} & {{- 0.0799} - {0.4928{\mathbb{i}}}} \\ {{- 0.3387} - {0.2385{\mathbb{i}}}} & {0.5212 + {0.1951{\mathbb{i}}}} & {{- 0.2917} + {0.1078{\mathbb{i}}}} \\ {{- 0.0962} + {0.2508{\mathbb{i}}}} & {{- 0.2418} + {0.1026{\mathbb{i}}}} & {{- 0.0231} - {0.7937{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},64} \right)} = \begin{matrix} {{- 0.4063} + {0.3394{\mathbb{i}}}} & {{- 0.2621} + {0.0798{\mathbb{i}}}} & {0.1421 + {0.0583{\mathbb{i}}}} \\ {0.2466 + {0.0922{\mathbb{i}}}} & {0.4783 - {0.5634{\mathbb{i}}}} & {{- 0.0799} - {0.4928{\mathbb{i}}}} \\ {{- 0.4080} - {0.5055{\mathbb{i}}}} & {0.5212 + {0.1951{\mathbb{i}}}} & {{- 0.2917} + {0.1078{\mathbb{i}}}} \\ {{- 0.4767} - {0.0347{\mathbb{i}}}} & {{- 0.2418} + {0.1026{\mathbb{i}}}} & {{- 0.0231} - {0.7937{\mathbb{i}}}} \end{matrix}$

(4) 6-Bit Rank 4 Codebook:

$\;{{{ans}\left( {\text{:},\text{:},1} \right)} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 & 0.5000 \\ 0.5000 & {0.0000 + {0.5000{\mathbb{i}}}} & {{- 0.5000} + {0.0000{\mathbb{i}}}} & {{- 0.0000} - {0.5000{\mathbb{i}}}} \\ 0.5000 & {{- 0.5000} + {0.0000{\mathbb{i}}}} & {0.5000 - {0.0000{\mathbb{i}}}} & {{- 0.5000} + {0.0000{\mathbb{i}}}} \\ 0.5000 & {{- 0.0000} - {0.5000{\mathbb{i}}}} & {{- 0.5000} + {0.0000{\mathbb{i}}}} & {0.0000 + {0.5000{\mathbb{i}}}} \end{matrix}}$ ${{ans}\left( {\text{:},\text{:},2} \right)} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 & 0.5000 \\ {0.3536 + {0.3536{\mathbb{i}}}} & {{- 0.3536} + {0.3536{\mathbb{i}}}} & {{- 0.3536} - {0.3536{\mathbb{i}}}} & {0.3536 - {0.3536{\mathbb{i}}}} \\ {0.0000 + {0.5000{\mathbb{i}}}} & {{- 0.0000} - {0.5000{\mathbb{i}}}} & {0.0000 + {0.5000{\mathbb{i}}}} & {{- 0.0000} - {0.5000{\mathbb{i}}}} \\ {{- 0.3536} + {0.3536{\mathbb{i}}}} & {0.3536 + {0.3536{\mathbb{i}}}} & {0.3536 - {0.3536{\mathbb{i}}}} & {{- 0.3536} - {0.3536{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},3} \right)} = \begin{matrix} {0.4619 + {0.1913{\mathbb{i}}}} & {{- 0.1913} + {0.4619{\mathbb{i}}}} & {{- 0.4619} - {0.1913{\mathbb{i}}}} & {0.1913 - {0.4619{\mathbb{i}}}} \\ {0.4455 - {0.2270{\mathbb{i}}}} & {0.4455 - {0.2270{\mathbb{i}}}} & {0.4455 - {0.2270{\mathbb{i}}}} & {0.4455 - {0.2270{\mathbb{i}}}} \\ {{- 0.1167} + {0.4862{\mathbb{i}}}} & {0.4862 + {0.1167{\mathbb{i}}}} & {0.1167 - {0.4862{\mathbb{i}}}} & {{- 0.4862} - {0.1167{\mathbb{i}}}} \\ {0.2939 + {0.4045{\mathbb{i}}}} & {{- 0.2939} - {0.4045{\mathbb{i}}}} & {0.2939 + {0.4045{\mathbb{i}}}} & {{- 0.2939} - {0.4045{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},4} \right)} = \begin{matrix} {- 0.5000} & {- 0.5000`} & {- 0.5000} & {- 0.5000} \\ {0.4985 - {0.0392{\mathbb{i}}}} & {0.0392 + {0.4985{\mathbb{i}}}} & {{- 0.4985} + {0.0392{\mathbb{i}}}} & {{- 0.0392} - {0.4985{\mathbb{i}}}} \\ {0.4938 - {0.0782{\mathbb{i}}}} & {{- 0.4938} + {0.0782{\mathbb{i}}}} & {0.4938 - {0.0782{\mathbb{i}}}} & {{- 0.4938} + {0.0782{\mathbb{i}}}} \\ {{- 0.4862} + {0.1167{\mathbb{i}}}} & {0.1167 + {0.4862{\mathbb{i}}}} & {0.4862 - {0.1167{\mathbb{i}}}} & {{- 0.1167} - {0.4862{\mathbb{i}}}} \end{matrix}$

$\;{{{ans}\left( {\text{:},\text{:},5} \right)} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 & 0.5000 \\ {0.4619 + {0.1913{\mathbb{i}}}} & {{- 0.1913} + {0.4619{\mathbb{i}}}} & {{- 0.4619} - {0.1913{\mathbb{i}}}} & {0.1913 - {0.4619{\mathbb{i}}}} \\ {0.3536 + {0.3536{\mathbb{i}}}} & {{- 0.3536} - {0.3536{\mathbb{i}}}} & {0.3536 + {0.3536{\mathbb{i}}}} & {{- 0.3536} - {0.3536{\mathbb{i}}}} \\ {0.1913 + {0.4619{\mathbb{i}}}} & {0.4619 - {0.1913{\mathbb{i}}}} & {{- 0.1913} - {0.4619{\mathbb{i}}}} & {{- 0.4619} + {0.1913{\mathbb{i}}}} \end{matrix}}$ ${{ans}\left( {\text{:},\text{:},6} \right)} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 & 0.5000 \\ {0.1913 + {0.4619{\mathbb{i}}}} & {{- 0.4619} + {0.1913`{\mathbb{i}}}} & {{- 0.1913} - {0.4619{\mathbb{i}}}} & {0.4619 - {0.1913{\mathbb{i}}}} \\ {{- 0.3536} + {0.3536{\mathbb{i}}}} & {0.3536 - {0.3536{\mathbb{i}}}} & {{- 0.3536} + {0.3536{\mathbb{i}}}} & {0.3536 - {0.3536{\mathbb{i}}}} \\ {{- 0.4619} - {0.1913{\mathbb{i}}}} & {{- 0.1913} + {0.4619{\mathbb{i}}}} & {0.4619 + {0.1913{\mathbb{i}}}} & {0.1913 - {0.4619{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},7} \right)} = \begin{matrix} {0.2437 + {0.4837{\mathbb{i}}}} & {0.4258 + {0.0076{\mathbb{i}}}} & {0.5210 - {0.4265{\mathbb{i}}}} & {{- 0.2364} - {0.1268{\mathbb{i}}}} \\ {0.5740 - {0.3102{\mathbb{i}}}} & {0.0596 + {0.3222{\mathbb{i}}}} & {0.2922 + {0.0454{\mathbb{i}}}} & {0.6147 + {0.0409{\mathbb{i}}}} \\ {0.2823 - {0.3783{\mathbb{i}}}} & {{- 0.4186} + {0.1136{\mathbb{i}}}} & {0.3628 + {0.1522{\mathbb{i}}}} & {{- 0.6500} - {0.1085{\mathbb{i}}}} \\ {0.2062 + {0.1248{\mathbb{i}}}} & {0.1513 + {0.7073{\mathbb{i}}}} & {{- 0.5498} - {0.0464{\mathbb{i}}}} & {{- 0.2340} - {0.2440{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},8} \right)} = \begin{matrix} {0.0591 - {0.4953{\mathbb{i}}}} & {0.0742 + {0.2714{\mathbb{i}}}} & {0.3370 + {0.2000{\mathbb{i}}}} & {{- 0.1041} - {0.7125{\mathbb{i}}}} \\ {0.5545 + {0.3908{\mathbb{i}}}} & {{- 0.0591} - {0.0963{\mathbb{i}}}} & {0.2158 + {0.4089{\mathbb{i}}}} & {{- 0.5589} + {0.0299{\mathbb{i}}}} \\ {0.2593 - {0.0234{\mathbb{i}}}} & {{- 0.5800} - {0.1645{\mathbb{i}}}} & {{- 0.2363} - {0.5965{\mathbb{i}}}} & {{- 0.2149} - {0.3331{\mathbb{i}}}} \\ {0.3300 + {0.3380{\mathbb{i}}}} & {0.4619 + {0.5755{\mathbb{i}}}} & {0.0175 - {0.4698{\mathbb{i}}}} & {0.0748 - {0.0748{\mathbb{i}}}} \end{matrix}$

${{ans}\left( {\text{:},\text{:},9} \right)} = \begin{matrix} {0.5019 - {0.2171{\mathbb{i}}}} & {0.2121 - {0.4391{\mathbb{i}}}} & {0.4649 + {0.1545{\mathbb{i}}}} & {{- 0.4709} + {0.0383{\mathbb{i}}}} \\ {0.0570 + {0.3132{\mathbb{i}}}} & {0.3637 + {0.1907{\mathbb{i}}}} & {0.5971 - {0.5122{\mathbb{i}}}} & {0.3285 + {0.0569{\mathbb{i}}}} \\ {0.2569 - {0.1797{\mathbb{i}}}} & {0.0539 - {0.3157{\mathbb{i}}}} & {{- 0.0025} + {0.3671{\mathbb{i}}}} & {0.8042 + {0.1327{\mathbb{i}}}} \\ {0.6949 + {0.1358{\mathbb{i}}}} & {{- 0.5835} + {0.3879{\mathbb{i}}}} & {0.0058 - {0.0797{\mathbb{i}}}} & {0.0341 + {0.0124{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},10} \right)} = \begin{matrix} {0.2199 + {0.3199{\mathbb{i}}}} & {0.6305 - {0.4345{\mathbb{i}}}} & {{- 0.0525} + {0.2357{\mathbb{i}}}} & {{- 0.3110} - {0.3286{\mathbb{i}}}} \\ {0.5983 + {0.1264{\mathbb{i}}}} & {{- 0.0683} + {0.2985{\mathbb{i}}}} & {0.5833 + {0.2614{\mathbb{i}}}} & {0.2969 - {0.1888{\mathbb{i}}}} \\ {{- 0.5576} - {0.2903{\mathbb{i}}}} & {{- 0.0495} - {0.3235{\mathbb{i}}}} & {0.3552 + {0.3750{\mathbb{i}}}} & {0.2888 - {0.3841{\mathbb{i}}}} \\ {{- 0.0940} - {0.2672{\mathbb{i}}}} & {{- 0.0481} + {0.4588{\mathbb{i}}}} & {0.0118 + {0.5160{\mathbb{i}}}} & {{- 0.6633} - {0.0260{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},11} \right)} = \begin{matrix} {0.4534 + {0.1075{\mathbb{i}}}} & {0.2632 + {0.4348{\mathbb{i}}}} & {0.1838 + {0.6429{\mathbb{i}}}} & {0.2307 + {0.1558{\mathbb{i}}}} \\ {{- 0.0163} + {0.7556{\mathbb{i}}}} & {0.3698 - {0.0222{\mathbb{i}}}} & {0.0055 - {0.3112{\mathbb{i}}}} & {0.3467 - {0.2728{\mathbb{i}}}} \\ {0.0681 + {0.3202{\mathbb{i}}}} & {{- 0.4066} - {0.6110{\mathbb{i}}}} & {0.3882 + {0.2198{\mathbb{i}}}} & {0.1705 + {0.3551{\mathbb{i}}}} \\ {0.1259 + {0.2977{\mathbb{i}}}} & {0.0347 + {0.2542{\mathbb{i}}}} & {0.4482 - {0.2369{\mathbb{i}}}} & {{- 0.7319} + {0.1924{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},12} \right)} = \begin{matrix} {0.0534 - {0.5633{\mathbb{i}}}} & {{- 0.4493} + {0.3172{\mathbb{i}}}} & {{- 0.3346} + {0.1479{\mathbb{i}}}} & {{- 0.4868} - {0.0811{\mathbb{i}}}} \\ {{- 0.1472} - {0.1184{\mathbb{i}}}} & {{- 0.5422} + {0.3203{\mathbb{i}}}} & {0.2302 - {0.4770{\mathbb{i}}}} & {0.4913 + {0.2140{\mathbb{i}}}} \\ {{- 0.7445} + {0.0533{\mathbb{i}}}} & {{- 0.1168} - {0.4073{\mathbb{i}}}} & {{- 0.1288} - {0.3642{\mathbb{i}}}} & {{- 0.2957} - {0.1634{\mathbb{i}}}} \\ {0.2881 - {0.0638{\mathbb{i}}}} & {0.3438 + {0.0562{\mathbb{i}}}} & {{- 0.1358} - {0.6465{\mathbb{i}}}} & {{- 0.3579} + {0.4766{\mathbb{i}}}} \end{matrix}$

${{ans}\left( {\text{:},\text{:},13} \right)} = \begin{matrix} {0.4192 + {0.3317{\mathbb{i}}}} & {0.6462 - {0.3049{\mathbb{i}}}} & {0.1765 - {0.0924{\mathbb{i}}}} & {0.3680 + {0.1691{\mathbb{i}}}} \\ {{- 0.2340} + {0.0529{\mathbb{i}}}} & {{- 0.0388} - {0.4789{\mathbb{i}}}} & {0.6212 - {0.0425{\mathbb{i}}}} & {{- 0.5223} + {0.2262{\mathbb{i}}}} \\ {0.0491 - {0.2199{\mathbb{i}}}} & {{- 0.2962} + {0.1142{\mathbb{i}}}} & {0.5130 + {0.5000{\mathbb{i}}}} & {0.5366 + {0.2175{\mathbb{i}}}} \\ {{- 0.7746} + {0.0772{\mathbb{i}}}} & {0.1089 - {0.3821{\mathbb{i}}}} & {{- 0.2395} + {0.0459{\mathbb{i}}}} & {0.4196 + {0.0255{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},14} \right)} = \begin{matrix} {0.0848 + {0.2248{\mathbb{i}}}} & {0.5883 + {0.1906{\mathbb{i}}}} & {{- 0.4550} - {0.1147{\mathbb{i}}}} & {{- 0.5202} + {0.2628{\mathbb{i}}}} \\ {{- 0.2010} + {0.7357{\mathbb{i}}}} & {{- 0.0481} - {0.2343{\mathbb{i}}}} & {{- 0.3923} + {0.0511{\mathbb{i}}}} & {0.3107 - {0.3288{\mathbb{i}}}} \\ {0.3862 + {0.2270{\mathbb{i}}}} & {0.2323 + {0.5375{\mathbb{i}}}} & {0.1654 - {0.1078{\mathbb{i}}}} & {0.6089 + {0.2161{\mathbb{i}}}} \\ {{- 0.3114} + {0.2509{\mathbb{i}}}} & {0.4661 + {0.0138{\mathbb{i}}}} & {0.6358 + {0.4245{\mathbb{i}}}} & {{- 0.1325} - {0.1438{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},15} \right)} = \begin{matrix} {0.3968 - {0.0671{\mathbb{i}}}} & {0.3110 + {0.1356{\mathbb{i}}}} & {{- 0.2168} - {0.4540{\mathbb{i}}}} & {{- 0.6008} - {0.3299{\mathbb{i}}}} \\ {{- 0.8289} - {0.0667{\mathbb{i}}}} & {{- 0.2015} + {0.0416{\mathbb{i}}}} & {0.1102 - {0.2871{\mathbb{i}}}} & {{- 0.4011} - {0.1031{\mathbb{i}}}} \\ {0.1770 - {0.1687{\mathbb{i}}}} & {{- 0.3325} - {0.1370{\mathbb{i}}}} & {0.0567 - {0.7634{\mathbb{i}}}} & {0.2785 + {0.3837{\mathbb{i}}}} \\ {{- 0.2814} + {0.0866{\mathbb{i}}}} & {0.7347 - {0.4165{\mathbb{i}}}} & {{- 0.0390} - {0.2544{\mathbb{i}}}} & {0.3519 - {0.1000{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},16} \right)} = \begin{matrix} {0.0743 - {0.7846{\mathbb{i}}}} & {{- 0.4063} + {0.3394{\mathbb{i}}}} & {{- 0.2621} + {0.0798{\mathbb{i}}}} & {0.1421 + {0.1583{\mathbb{i}}}} \\ {0.1307 - {0.3436{\mathbb{i}}}} & {0.2466 + {0.0922{\mathbb{i}}}} & {0.4783 - {0.5634{\mathbb{i}}}} & {{- 0.0799} - {0.4928{\mathbb{i}}}} \\ {{- 0.3387} - {0.2385{\mathbb{i}}}} & {{- 0.4080} - {0.5055{\mathbb{i}}}} & {0.5212 + {0.1951{\mathbb{i}}}} & {{- 0.2917} + {0.1078{\mathbb{i}}}} \\ {{- 0.0962} + {0.2508{\mathbb{i}}}} & {{- 0.4767} - {0.0347{\mathbb{i}}}} & {{- 0.2418} + {0.1026{\mathbb{i}}}} & {{- 0.0231} - {0.7937{\mathbb{i}}}} \end{matrix}$

2. 4-Bit Codebook

(1) 4-bit Rank 1 Codebook:

${{ans}\left( {\text{:},\text{:},1} \right)} = \begin{matrix} 0.5000 \\ 0.5000 \\ 0.5000 \\ 0.5000 \end{matrix}$ ${{ans}\left( {\text{:},\text{:},2} \right)} = \begin{matrix} 0.5000 \\ {0.0000 + {0.5000{\mathbb{i}}}} \\ {{- 0.5000} + {0.0000{\mathbb{i}}}} \\ {{- 0.0000} - {0.5000{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},3} \right)} = \begin{matrix} 0.5000 \\ {{- 0.5000} + {0.0000{\mathbb{i}}}} \\ {0.5000 - {0.0000{\mathbb{i}}}} \\ {{- 0.5000} + {0.0000{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},4} \right)} = \begin{matrix} 0.5000 \\ {{- 0.0000} - {0.5000{\mathbb{i}}}} \\ {{- 0.5000} + {0.0000{\mathbb{i}}}} \\ {0.0000 + {0.5000{\mathbb{i}}}} \end{matrix}$

${{ans}\left( {\text{:},\text{:},5} \right)} = \begin{matrix} 0.5000 \\ {0.3536 + {0.3536{\mathbb{i}}}} \\ {0.0000 + {0.5000{\mathbb{i}}}} \\ {{- 0.3536} + {0.3536{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},6} \right)} = \begin{matrix} 0.5000 \\ {{- 0.3536} + {0.3536{\mathbb{i}}}} \\ {{- 0.0000} - {0.5000{\mathbb{i}}}} \\ {0.3536 + {0.3536{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},7} \right)} = \begin{matrix} 0.5000 \\ {{- 0.3536} - {0.3536{\mathbb{i}}}} \\ {0.0000 + {0.5000{\mathbb{i}}}} \\ {0.3536 - {0.3536{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},8} \right)} = \begin{matrix} 0.5000 \\ {0.3536 - {0.3536{\mathbb{i}}}} \\ {{- 0.0000} - {0.5000{\mathbb{i}}}} \\ {{- 0.3536} - {0.3536{\mathbb{i}}}} \end{matrix}$

${{ans}\left( {\text{:},\text{:},9} \right)} = \begin{matrix} {0.4619 + {0.1913{\mathbb{i}}}} \\ {0.4455 - {0.2270{\mathbb{i}}}} \\ {{- 0.1167} + {0.4862{\mathbb{i}}}} \\ {0.2939 + {0.4045{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},10} \right)} = \begin{matrix} {{- 0.1913} + {0.4619{\mathbb{i}}}} \\ {0.4455 - {0.2270{\mathbb{i}}}} \\ {0.4862 + {0.1167{\mathbb{i}}}} \\ {{- 0.2939} - {0.4045{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},11} \right)} = \begin{matrix} {{- 0.4619} - {0.1913{\mathbb{i}}}} \\ {0.4455 - {0.2270{\mathbb{i}}}} \\ {0.1167 - {0.4862{\mathbb{i}}}} \\ {0.2939 + {0.4045{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},12} \right)} = \begin{matrix} {0.1913 - {0.4619{\mathbb{i}}}} \\ {0.4455 - {0.2270{\mathbb{i}}}} \\ {{- 0.4862} - {0.1167{\mathbb{i}}}} \\ {{- 0.2939} - {0.4045{\mathbb{i}}}} \end{matrix}$

${{ans}\left( {\text{:},\text{:},13} \right)} = \begin{matrix} {- 0.5000} \\ {0.4985 - {0.0392{\mathbb{i}}}} \\ {0.4938 - {0.0782{\mathbb{i}}}} \\ {{- 0.4862} + {0.1167{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},14} \right)} = \begin{matrix} {- 0.5000} \\ {0.0392 + {0.4985{\mathbb{i}}}} \\ {{- 0.4938} + {0.0782{\mathbb{i}}}} \\ {0.1167 + {0.4862{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},15} \right)} = \begin{matrix} {- 0.5000} \\ {{- 0.4985} + {0.0392{\mathbb{i}}}} \\ {0.4938 - {0.0782{\mathbb{i}}}} \\ {0.4862 - {0.1167{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},16} \right)} = \begin{matrix} {- 0.5000} \\ {{- 0.0392} - {0.4985{\mathbb{i}}}} \\ {{- 0.4938} + {0.0782{\mathbb{i}}}} \\ {{- 0.1167} - {0.4862{\mathbb{i}}}} \end{matrix}$

(2) 4-Bit Rank 2 Codebook:

${{ans}\left( {\text{:},\text{:},1} \right)} = \begin{matrix} 0.5000 & 0.5000 \\ 0.5000 & {0.0000 + {0.5000{\mathbb{i}}}} \\ 0.5000 & {{- 0.5000} + {0.0000{\mathbb{i}}}} \\ 0.5000 & {{- 0.0000} - {0.5000{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},2} \right)} = \begin{matrix} 0.5000 & 0.5000 \\ 0.5000 & {{- 0.5000} + {0.0000{\mathbb{i}}}} \\ 0.5000 & {0.5000 - {0.0000{\mathbb{i}}}} \\ 0.5000 & {{- 0.5000} + {0.0000{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},3} \right)} = \begin{matrix} 0.5000 & 0.5000 \\ 0.5000 & {{- 0.0000} - {0.5000{\mathbb{i}}}} \\ 0.5000 & {{- 0.5000} + {0.0000{\mathbb{i}}}} \\ 0.5000 & {0.0000 + {0.5000{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},4} \right)} = \begin{matrix} 0.5000 & 0.5000 \\ {0.0000 + {0.5000{\mathbb{i}}}} & {{- 0.5000} + {0.0000{\mathbb{i}}}} \\ {{- 0.5000} + {0.0000{\mathbb{i}}}} & {0.5000 - {0.0000{\mathbb{i}}}} \\ {{- 0.0000} - {0.5000{\mathbb{i}}}} & {{- 0.5000} + {0.0000{\mathbb{i}}}} \end{matrix}$

${{ans}\left( {\text{:},\text{:},5} \right)} = \begin{matrix} 0.5000 & 0.5000 \\ {0.0000 + {0.5000{\mathbb{i}}}} & {{- 0.0000} - {0.5000{\mathbb{i}}}} \\ {{- 0.5000} + {0.0000{\mathbb{i}}}} & {{- 0.5000} + {0.0000{\mathbb{i}}}} \\ {{- 0.0000} - {0.5000{\mathbb{i}}}} & {0.0000 + {0.5000{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},6} \right)} = \begin{matrix} 0.5000 & 0.5000 \\ {{- 0.5000} + {0.0000{\mathbb{i}}}} & {{- 0.0000} - {0.5000{\mathbb{i}}}} \\ {0.5000 - {0.0000{\mathbb{i}}}} & {{- 0.5000} + {0.0000{\mathbb{i}}}} \\ {{- 0.5000} + {0.0000{\mathbb{i}}}} & {0.0000 + {0.5000{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},7} \right)} = \begin{matrix} 0.5000 & 0.5000 \\ {0.3536 + {0.3536{\mathbb{i}}}} & {{- 0.3536} + {0.3536{\mathbb{i}}}} \\ {0.0000 + {0.5000{\mathbb{i}}}} & {{- 0.0000} - {0.5000{\mathbb{i}}}} \\ {{- 0.3536} + {0.3536{\mathbb{i}}}} & {0.3536 + {0.3536{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},8} \right)} = \begin{matrix} 0.5000 & 0.5000 \\ {0.3536 + {0.3536{\mathbb{i}}}} & {{- 0.3536} - {0.3536{\mathbb{i}}}} \\ {0.0000 + {0.5000{\mathbb{i}}}} & {0.0000 + {0.5000{\mathbb{i}}}} \\ {{- 0.3536} + {0.3536{\mathbb{i}}}} & {0.3536 - {0.3536{\mathbb{i}}}} \end{matrix}$

${{ans}\left( {\text{:},\text{:},9} \right)} = \begin{matrix} 0.5000 & 0.5000 \\ {0.3536 + {0.3536{\mathbb{i}}}} & {0.3536 - {0.3536{\mathbb{i}}}} \\ {0.0000 + {0.5000{\mathbb{i}}}} & {{- 0.0000} - {0.5000{\mathbb{i}}}} \\ {{- 0.3536} + {0.3536{\mathbb{i}}}} & {{- 0.3536} - {0.3536{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},10} \right)} = \begin{matrix} 0.5000 & 0.5000 \\ {{- 0.3536} + {0.3536{\mathbb{i}}}} & {{- 0.3536} - {0.3536{\mathbb{i}}}} \\ {{- 0.0000} - {0.5000{\mathbb{i}}}} & {0.0000 + {0.5000{\mathbb{i}}}} \\ {0.3536 + {0.3536{\mathbb{i}}}} & {0.3536 - {0.3536{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},11} \right)} = \begin{matrix} 0.5000 & 0.5000 \\ {{- 0.3536} + {0.3536{\mathbb{i}}}} & {0.3536 - {0.3536{\mathbb{i}}}} \\ {{- 0.0000} - {0.5000{\mathbb{i}}}} & {{- 0.0000} - {0.5000{\mathbb{i}}}} \\ {0.3536 + {0.3536{\mathbb{i}}}} & {{- 0.3536} - {0.3536{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},12} \right)} = \begin{matrix} 0.5000 & 0.5000 \\ {{- 0.3536} - {0.3536{\mathbb{i}}}} & {0.3536 - {0.3536{\mathbb{i}}}} \\ {0.0000 + {0.5000{\mathbb{i}}}} & {{- 0.0000} - {0.5000{\mathbb{i}}}} \\ {0.3536 - {0.3536{\mathbb{i}}}} & {{- 0.3536} - {0.3536{\mathbb{i}}}} \end{matrix}$

${{ans}\left( {\text{:},\text{:},13} \right)} = \begin{matrix} {0.4619 + {0.1913{\mathbb{i}}}} & {0.1913 - {0.4619{\mathbb{i}}}} \\ {0.4455 - {0.2270{\mathbb{i}}}} & {0.4455 - {0.2270{\mathbb{i}}}} \\ {{- 0.1167} + {0.4862{\mathbb{i}}}} & {{- 0.4862} - {0.1167{\mathbb{i}}}} \\ {0.2939 + {0.4045{\mathbb{i}}}} & {{- 0.2939} - {0.4045{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},14} \right)} = \begin{matrix} {{- 0.1913} + {0.4619{\mathbb{i}}}} & {{- 0.4619} - {0.1913{\mathbb{i}}}} \\ {0.4455 - {0.2270{\mathbb{i}}}} & {0.4455 - {0.2270{\mathbb{i}}}} \\ {0.4862 + {0.1167{\mathbb{i}}}} & {0.1167 - {0.4862{\mathbb{i}}}} \\ {{- 0.2939} - {0.4045{\mathbb{i}}}} & {0.2939 + {0.4045{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},15} \right)} = \begin{matrix} {- 0.5000} & {- 0.5000} \\ {0.4985 - {0.0392{\mathbb{i}}}} & {0.0392 + {0.4985{\mathbb{i}}}} \\ {0.4938 - {0.0782{\mathbb{i}}}} & {{- 0.4938} + {0.0782{\mathbb{i}}}} \\ {{- 0.4862} + {0.1167{\mathbb{i}}}} & {0.1167 + {0.4862{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},16} \right)} = \begin{matrix} {- 0.5000} & {- 0.5000} \\ {0.4985 - {0.0392{\mathbb{i}}}} & {{- 0.0392} - {0.4985{\mathbb{i}}}} \\ {0.4938 - {0.0782{\mathbb{i}}}} & {{- 0.4938} + {0.0782{\mathbb{i}}}} \\ {{- 0.4862} + {0.1167{\mathbb{i}}}} & {{- 0.1167} - {0.4862{\mathbb{i}}}} \end{matrix}$

(3) 4-Bit Rank 3 Codebook:

${{ans}\left( {\text{:},\text{:},1} \right)} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 \\ 0.5000 & {0.0000 + {0.5000{\mathbb{i}}}} & {{- 0.5000} + {0.0000{\mathbb{i}}}} \\ 0.5000 & {{- 0.5000} + {0.0000{\mathbb{i}}}} & {0.5000 - {0.0000{\mathbb{i}}}} \\ 0.5000 & {{- 0.0000} - {0.5000{\mathbb{i}}}} & {{- 0.5000} + {0.0000{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},2} \right)} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 \\ 0.5000 & {0.0000 + {0.5000{\mathbb{i}}}} & {{- 0.0000} - {0.5000{\mathbb{i}}}} \\ 0.5000 & {{- 0.5000} + {0.0000{\mathbb{i}}}} & {{- 0.5000} + {0.0000{\mathbb{i}}}} \\ 0.5000 & {{- 0.0000} - {0.5000{\mathbb{i}}}} & {0.0000 + {0.5000{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},3} \right)} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 \\ 0.5000 & {{- 0.5000} + {0.0000{\mathbb{i}}}} & {{- 0.0000} - {0.5000{\mathbb{i}}}} \\ 0.5000 & {0.5000 - {0.0000{\mathbb{i}}}} & {{- 0.5000} + {0.0000{\mathbb{i}}}} \\ 0.5000 & {{- 0.5000} + {0.0000{\mathbb{i}}}} & {0.0000 + {0.5000{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:}, 4} \right)} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 \\ {0.0000 + {0.5000{\mathbb{i}}}} & {{- 0.5000} + {0.0000{\mathbb{i}}}} & {{- 0.0000} - {0.5000{\mathbb{i}}}} \\ {{- 0.5000} + {0.0000{\mathbb{i}}}} & {0.5000 - {0.0000{\mathbb{i}}}} & {{- 0.5000} + {0.0000{\mathbb{i}}}} \\ {{- 0.0000} - {0.5000{\mathbb{i}}}} & {{- 0.5000} + {0.0000{\mathbb{i}}}} & {0.0000 + {0.5000{\mathbb{i}}}} \end{matrix}$

${{ans}\left( {\text{:},\text{:},5} \right)} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 \\ {0.3536 + {0.3536{\mathbb{i}}}} & {{- 0.3536} + {0.3536{\mathbb{i}}}} & {{- 0.3536} - {0.3536{\mathbb{i}}}} \\ {0.0000 + {0.5000{\mathbb{i}}}} & {{- 0.0000} - {0.5000{\mathbb{i}}}} & {0.0000 + {0.5000{\mathbb{i}}}} \\ {{- 0.3536} + {0.3536{\mathbb{i}}}} & {0.3536 + {0.3536{\mathbb{i}}}} & {0.3536 - {0.3536{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:}, 6} \right)} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 \\ {0.3536 + {0.3536{\mathbb{i}}}} & {{- 0.3536} + {0.3536{\mathbb{i}}}} & {0.3536 - {0.3536{\mathbb{i}}}} \\ {0.0000 + {0.5000{\mathbb{i}}}} & {{- 0.0000} - {0.5000{\mathbb{i}}}} & {{- 0.0000} - {0.5000{\mathbb{i}}}} \\ {{- 0.3536} + {0.3536{\mathbb{i}}}} & {0.3536 + {0.3536{\mathbb{i}}}} & {{- 0.3536} - {0.3536{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:}, 7} \right)} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 \\ {0.3536 + {0.3536{\mathbb{i}}}} & {{- 0.3536} - {0.3536{\mathbb{i}}}} & {0.3536 - {0.3536{\mathbb{i}}}} \\ {0.0000 + {0.5000{\mathbb{i}}}} & {0.0000 + {0.5000{\mathbb{i}}}} & {{- 0.0000} - {0.5000{\mathbb{i}}}} \\ {{- 0.3536} + {0.3536{\mathbb{i}}}} & {0.3536 - {0.3536{\mathbb{i}}}} & {{- 0.3536} - {0.3536{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:}, 8} \right)} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 \\ {{- 0.3536} + {0.3536{\mathbb{i}}}} & {{- 0.3536} - {0.3536{\mathbb{i}}}} & {0.3536 - {0.3536{\mathbb{i}}}} \\ {{- 0.0000} - {0.5000{\mathbb{i}}}} & {0.0000 + {0.5000{\mathbb{i}}}} & {{- 0.0000} - {0.5000{\mathbb{i}}}} \\ {0.3536 + {0.3536{\mathbb{i}}}} & {0.3536 - {0.3536{\mathbb{i}}}} & {{- 0.3536} - {0.3536{\mathbb{i}}}} \end{matrix}$

${{ans}\left( {\text{:},\text{:},9} \right)} = \begin{matrix} {0.4619 + {0.1913{\mathbb{i}}}} & {{- 0.1913} + {0.4619{\mathbb{i}}}} & {{- 0.4619} - {0.1913{\mathbb{i}}}} \\ {0.4455 - {0.2270{\mathbb{i}}}} & {0.4455 - {0.2270{\mathbb{i}}}} & {0.4455 - {0.2270{\mathbb{i}}}} \\ {{- 0.1167} + {0.4862{\mathbb{i}}}} & {0.4862 + {0.1167{\mathbb{i}}}} & {0.1167 - {0.4862{\mathbb{i}}}} \\ {0.2939 + {0.4045{\mathbb{i}}}} & {{- 0.2939} - {0.4045{\mathbb{i}}}} & {0.2939 + {0.4045{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},10} \right)} = \begin{matrix} {0.4619 + {0.1913{\mathbb{i}}}} & {{- 0.1913} + {0.4619{\mathbb{i}}}} & {0.1913 - {0.4619{\mathbb{i}}}} \\ {0.4455 - {0.2270{\mathbb{i}}}} & {0.4455 - {0.2270{\mathbb{i}}}} & {0.4455 - {0.2270{\mathbb{i}}}} \\ {{- 0.1167} + {0.4862{\mathbb{i}}}} & {0.4862 + {0.1167{\mathbb{i}}}} & {{- 0.4862} - {0.1167{\mathbb{i}}}} \\ {0.2939 + {0.4045{\mathbb{i}}}} & {{- 0.2939} - {0.4045{\mathbb{i}}}} & {{- 0.2939} - {0.4045{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},11} \right)} = \begin{matrix} {0.4619 + {0.1913{\mathbb{i}}}} & {{- 0.4619} - {0.1913{\mathbb{i}}}} & {0.1913 - {0.4619{\mathbb{i}}}} \\ {0.4455 - {0.2270{\mathbb{i}}}} & {0.4455 - {0.2270{\mathbb{i}}}} & {0.4455 - {0.2270{\mathbb{i}}}} \\ {{- 0.1167} + {0.4862{\mathbb{i}}}} & {0.1167 - {0.4862{\mathbb{i}}}} & {{- 0.4862} - {0.1167{\mathbb{i}}}} \\ {0.2939 + {0.4045{\mathbb{i}}}} & {0.2939 + {0.4045{\mathbb{i}}}} & {{- 0.2939} - {0.4045{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},12} \right)} = \begin{matrix} {{- 0.1913} + {0.4619{\mathbb{i}}}} & {{- 0.4619} - {0.1913{\mathbb{i}}}} & {0.1913 - {0.4619{\mathbb{i}}}} \\ {0.4455 - {0.2270{\mathbb{i}}}} & {0.4455 - {0.2270{\mathbb{i}}}} & {0.4455 - {0.2270{\mathbb{i}}}} \\ {0.4862 + {0.1167{\mathbb{i}}}} & {0.1167 - {0.4862{\mathbb{i}}}} & {{- 0.4862} - {0.1167{\mathbb{i}}}} \\ {{- 0.2939} - {0.4045{\mathbb{i}}}} & {0.2939 + {0.4045{\mathbb{i}}}} & {{- 0.2939} - {0.4045{\mathbb{i}}}} \end{matrix}$

${{ans}\left( {\text{:},\text{:},13} \right)} = \begin{matrix} {- 0.5000} & {- 0.5000} & {- 0.5000} \\ {0.4985 - {0.0392{\mathbb{i}}}} & {0.0392 + {0.4985{\mathbb{i}}}} & {{- 0.4985} + {0.0392{\mathbb{i}}}} \\ {0.4938 - {0.0782{\mathbb{i}}}} & {{- 0.4938} + {0.0782{\mathbb{i}}}} & {0.4938 - {0.0782{\mathbb{i}}}} \\ {{- 0.4862} + {0.1167{\mathbb{i}}}} & {0.1167 + {0.4862{\mathbb{i}}}} & {0.4862 - {0.1167{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},14} \right)} = \begin{matrix} {- 0.5000} & {- 0.5000} & {- 0.5000} \\ {0.4985 - {0.0392{\mathbb{i}}}} & {0.0392 + {0.4985{\mathbb{i}}}} & {{- 0.0392} - {0.4985{\mathbb{i}}}} \\ {0.4938 - {0.0782{\mathbb{i}}}} & {{- 0.4938} + {0.0782{\mathbb{i}}}} & {{- 0.4938} + {0.0782{\mathbb{i}}}} \\ {{- 0.4862} + {0.1167{\mathbb{i}}}} & {0.1167 + {0.4862{\mathbb{i}}}} & {{- 0.1167} - {0.4862{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},15} \right)} = \begin{matrix} {- 0.5000} & {- 0.5000} & {- 0.5000} \\ {0.4985 - {0.0392{\mathbb{i}}}} & {{- 0.4985} + {0.0392{\mathbb{i}}}} & {{- 0.0392} - {0.4985{\mathbb{i}}}} \\ {0.4938 - {0.0782{\mathbb{i}}}} & {0.4938 - {0.0782{\mathbb{i}}}} & {{- 0.4938} + {0.0782{\mathbb{i}}}} \\ {{- 0.4862} + {0.1167{\mathbb{i}}}} & {0.4862 - {0.1167{\mathbb{i}}}} & {{- 0.1167} - {0.4862{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},16} \right)} = \begin{matrix} {- 0.5000} & {- 0.5000} & {- 0.5000} \\ {0.0392 + {0.4985{\mathbb{i}}}} & {{- 0.4985} + {0.0392{\mathbb{i}}}} & {{- 0.0392} - {0.4985{\mathbb{i}}}} \\ {{- 0.4938} - {0.0782{\mathbb{i}}}} & {0.4938 - {0.0782{\mathbb{i}}}} & {{- 0.4938} + {0.0782{\mathbb{i}}}} \\ {0.1167 + {0.4862{\mathbb{i}}}} & {0.4862 - {0.1167{\mathbb{i}}}} & {{- 0.1167} - {0.4862{\mathbb{i}}}} \end{matrix}$

(4) 4-Bit Rank 4 Codebook:

${{ans}\left( {\text{:},\text{:},1} \right)} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 & 0.5000 \\ 0.5000 & {0.0000 + {0.5000{\mathbb{i}}}} & {{- 0.5000} + {0.0000{\mathbb{i}}}} & {{- 0.0000} - {0.5000{\mathbb{i}}}} \\ 0.5000 & {{- 0.5000} + {0.0000{\mathbb{i}}}} & {0.5000 - {0.0000{\mathbb{i}}}} & {{- 0.5000} + {0.0000{\mathbb{i}}}} \\ 0.5000 & {{- 0.0000} - {0.5000{\mathbb{i}}}} & {{- 0.5000} + {0.0000{\mathbb{i}}}} & {0.0000 + {0.5000{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},2} \right)} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 & 0.5000 \\ {0.3536 + {0.3536{\mathbb{i}}}} & {{- 0.3536} + {0.3536{\mathbb{i}}}} & {{- 0.3536} - {0.3536{\mathbb{i}}}} & {0.3536 - {0.3536{\mathbb{i}}}} \\ {0.0000 + {0.5000{\mathbb{i}}}} & {{- 0.0000} - {0.5000{\mathbb{i}}}} & {0.0000 + {0.5000{\mathbb{i}}}} & {{- 0.0000} - {0.5000{\mathbb{i}}}} \\ {{- 0.3536} + {0.3536{\mathbb{i}}}} & {0.3536 + {0.3536{\mathbb{i}}}} & {0.3536 - {0.3536{\mathbb{i}}}} & {{- 0.3536} - {0.3536{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},3} \right)} = \begin{matrix} {0.4619 + {0.1913{\mathbb{i}}}} & {{- 0.1913} + {0.4619{\mathbb{i}}}} & {{- 0.4619} - {0.1913{\mathbb{i}}}} & {0.1913 - {0.4619{\mathbb{i}}}} \\ {0.4455 - {0.2270{\mathbb{i}}}} & {0.4455 - {0.2270{\mathbb{i}}}} & {0.4455 - {0.2270{\mathbb{i}}}} & {0.4455 - {0.2270{\mathbb{i}}}} \\ {{- 0.1167} + {0.4862{\mathbb{i}}}} & {0.4862 + {0.1167{\mathbb{i}}}} & {0.1167 - {0.4862{\mathbb{i}}}} & {{- 0.4862} - {0.1167{\mathbb{i}}}} \\ {0.2939 + {0.4045{\mathbb{i}}}} & {{- 0.2939} - {0.4045{\mathbb{i}}}} & {0.2939 + {0.4045{\mathbb{i}}}} & {{- 0.2939} - {0.4045{\mathbb{i}}}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},4} \right)} = \begin{matrix} {- 0.5000} & {- 0.5000} & {- 0.5000} & {- 0.5000} \\ {0.4985 - {0.0392{\mathbb{i}}}} & {0.0392 + {0.4985{\mathbb{i}}}} & {{- 0.4985} + {0.0392{\mathbb{i}}}} & {{- 0.0392} - {0.4985{\mathbb{i}}}} \\ {0.4938 - {0.0782{\mathbb{i}}}} & {{- 0.4938} + {0.0782{\mathbb{i}}}} & {0.4938 - {0.0782{\mathbb{i}}}} & {{- 0.4938} + {0.0782{\mathbb{i}}}} \\ {{- 0.4862} + {0.1167{\mathbb{i}}}} & {0.1167 + {0.4862{\mathbb{i}}}} & {0.4862 - {0.1167{\mathbb{i}}}} & {{- 0.1167} - {0.4862{\mathbb{i}}}} \end{matrix}$

FIG. 3 is a flowchart illustrating an exemplary MIMO communication method. The exemplary method may be performed by a base station and/or a terminal in a MIMO communication network. In 310, the method comprises storing a codebook in memory. In 320, the method recognizes a channel state of the MIMO communication network. In 330, the method determines a transmission rank. In 340, the method comprises determining a precoding matrix. And in 350, the method comprises performing precoding.

The methods described above may be recorded, stored, or fixed in one or more computer-readable storage media that includes program instructions to be implemented by a computer to cause a processor to execute or perform the program instructions. The media may also include, alone or in combination with the program instructions, data files, data structures, and the like. The media and program instructions may be those specially designed and constructed, or they may be of the kind well-known and available to those having skill in the computer software arts. Examples of computer-readable media include magnetic media such as hard disks, floppy disks, and magnetic tape; optical media such as CD ROM disks and DVD; magneto-optical media such as optical disks; and hardware devices that are specially configured to store and perform program instructions, such as read-only memory (ROM), random access memory (RAM), flash memory, and the like. Examples of program instructions include both machine code, such as produced by a compiler, and files containing higher level code that may be executed by the computer using an interpreter. The described hardware devices may be configured to act as one or more software modules in order to perform the operations and methods described above, or vice versa. In addition, a computer-readable storage medium may be distributed among computer systems connected through a network and computer-readable instructions or codes may be stored and executed in a decentralized manner.

FIG. 4 illustrates a configuration of a base station 410 and a terminal 420.

The base station 410 includes a memory 411, a processor 412, and a precoder 413. The terminal 420 includes a memory 421, a channel estimator 422, a processor 423, and a feedback unit 424.

The aforementioned codebooks may be stored in the memory 411 of the base station 410 and/or the memory 421 of the terminal 420. For example, any of the codebooks described herein may be stored in the memory 411 of the base station 410 and/or the memory 421 of the terminal 420.

The channel estimator 422 may estimate a channel between the base station 410 and the terminal 420. The processor 423 may select a preferred codeword matrix from the codebook stored in the memory 421, based on the estimated channel, and generate feedback data associated with the preferred codeword matrix. The feedback unit 424 may feed back the feedback data to the base station 410.

The base station 410 may receive the feedback data. The processor 412 may verify the preferred codeword matrix using the codebook stored in the memory 411. The precoder 413 may precode at least one data stream using the determined precoding matrix.

A number of exemplary embodiments have been described above. Nevertheless, it will be understood that various modifications may be made. For example, suitable results may be achieved if the described techniques are performed in a different order and/or if components in a described system, architecture, device, or circuit are combined in a different manner and/or replaced or supplemented by other components or their equivalents. Accordingly, other implementations are within the scope of the following claims. 

1. A terminal in a multiple input multiple output (MIMO) communication system, the terminal comprising: a memory configured to store a 6-bit codebook comprising 64 codeword matrices; a channel estimator configured to estimate a channel formed between a base station and the terminal; a processor configured to select a preferred codeword vector from the codebook stored in the memory; and a feedback unit configured to feed back feedback data associated with the preferred codeword vector to the base station, wherein the 6-bit codebook comprises columns 1 through 64: columns 1 through 4 0.5000 0.3260 + 0.6774i 0.1499 + 0.0347i 0.5000 0.3254 + 0.1709i 0.5009 + 0.3071i 0.5000 0.3254 + 0.1709i 0.1505 + 0.5412i 0.5000 −0.0250 + 0.4051i   0.1505 + 0.5412i 0.5000 0.0918 + 0.3270i 0.5000 0.1311 + 0.6387i 0.5000 0.2473 + 0.0541i 0.5000 0.4815 + 0.4045i columns 5 through 8 0.5000 0.0918 + 0.3270i 0.3841 + 0.3851i −0.5000   −0.1311 − 0.6387i   0.0056 − 0.3076i 0.5000 0.2473 + 0.0541i 0.2285 + 0.6580i −0.5000   −0.4815 − 0.4045i   −0.3448 − 0.0735i   0.5000 0.3260 + 0.6774i −0.5000   −0.3254 − 0.1709i   0.5000 0.3254 + 0.1709i −0.5000   0.0250 − 0.4051i columns 9 through 12 −0.5000   −0.0918 − 0.3270i   −0.0337 − 0.6193i   −0.5000   −0.2473 − 0.0541i   −0.4621 − 0.5019i   0.5000 0.1311 + 0.6387i 0.2280 + 0.1515i 0.5000 0.4815 + 0.4045i 0.2280 + 0.1515i −0.5000   −0.4422 − 0.0928i   −0.5000   −0.2479 − 0.5606i   0.5000 0.2479 + 0.5606i 0.5000 0.0137 + 0.2102i columns 13 through 16 −0.5000   −0.4422 − 0.0928i   −0.3841 − 0.3851i   0.5000 0.2479 + 0.5606i −0.0056 + 0.3076i   0.5000 0.2479 + 0.5606i 0.3448 + 0.0735i −0.5000   −0.0137 − 0.2102i   −0.2285 − 0.6580i   −0.5000   −0.0918 − 0.3270i   0.5000 0.2473 + 0.0541i 0.5000 0.1311 + 0.6387i −0.5000   −0.4815 − 0.4045i   columns 17 through 20 0.5000 0.3841 + 0.3851i 0.0918 + 0.3270i    0 + 0.5000i −0.3076 − 0.0056i   −0.0541 + 0.2473i   0.5000 0.3448 + 0.0735i 0.1311 + 0.6387i    0 + 0.5000i −0.6580 + 0.2285i   −0.4045 + 0.4815i   0.5000 0.0337 + 0.6193i    0 + 0.5000i −0.5019 + 0.4621i   0.5000 0.2280 + 0.1515i    0 + 0.5000i −0.1515 + 0.2280i   columns 21 through 24 0.5000 0.0337 + 0.6193i 0.4422 + 0.0928i    0 − 0.5000i 0.5019 − 0.4621i 0.5606 − 0.2479i 0.5000 0.2280 + 0.1515i 0.2479 + 0.5606i    0 − 0.5000i 0.1515 − 0.2280i 0.2102 − 0.0137i 0.5000 0.3841 + 0.3851i    0 − 0.5000i 0.3076 + 0.0056i 0.5000 0.3448 + 0.0735i    0 − 0.5000i 0.6580 − 0.2285i columns 25 through 28 −0.5000   −0.3841 − 0.3851i   −0.3260 − 0.6774i      0 − 0.5000i 0.3076 + 0.0056i 0.1709 − 0.3254i 0.5000 0.2285 + 0.6580i 0.3254 + 0.1709i    0 + 0.5000i −0.0735 + 0.3448i   −0.4051 − 0.0250i   −0.5000   −0.1499 − 0.0347i      0 − 0.5000i 0.3071 − 0.5009i 0.5000 0.1505 + 0.5412i    0 + 0.5000i −0.5412 + 0.1505i   columns 29 through 32 −0.5000   −0.1499 − 0.0347i   −0.0918 − 0.3270i      0 + 0.5000i −0.3071 + 0.5009i   −0.6387 + 0.1311i   0.5000 0.1505 + 0.5412i 0.2473 + 0.0541i    0 − 0.5000i 0.5412 − 0.1505i 0.4045 − 0.4815i −0.5000   −0.3841 − 0.3851i      0 + 0.5000i −0.3076 − 0.0056i   0.5000 0.2285 + 0.6580i    0 − 0.5000i 0.0735 − 0.3448i columns 33 through 36 0.5000 0.0337 + 0.6193i 0.0918 + 0.3270i 0.5000 0.2280 + 0.1515i 0.4815 + 0.4045i 0.5000 0.2280 + 0.1515i 0.1311 + 0.6387i −0.5000   −0.4621 − 0.5019i   −0.2473 − 0.0541i   0.5000 0.3841 + 0.3851i 0.5000 0.2285 + 0.6580i 0.5000 0.3448 + 0.0735i −0.5000   0.0056 − 0.3076i columns 37 through 40 0.5000 0.4422 + 0.0928i 0.0337 + 0.6193i    0 + 0.5000i −0.2102 + 0.0137i   −0.1515 + 0.2280i   −0.5000   −0.2479 − 0.5606i   −0.2280 − 0.1515i      0 + 0.5000i −0.5606 + 0.2479i   −0.5019 + 0.4621i   0.5000 0.0918 + 0.3270i    0 + 0.5000i −0.4045 + 0.4815i   −0.5000   −0.1311 − 0.6387i      0 + 0.5000i −0.0541 + 0.2473i   columns 41 through 44 0.5000 0.3841 + 0.3851i 0.4422 + 0.0928i −0.5000   −0.2285 − 0.6580i   −0.0137 − 0.2102i   0.5000 0.3448 + 0.0735i 0.2479 + 0.5606i 0.5000 −0.0056 + 0.3076i   0.2479 + 0.5606i 0.5000 0.0337 + 0.6193i −0.5000   −0.2280 − 0.1515i   0.5000 0.2280 + 0.1515i 0.5000 0.4621 + 0.5019i columns 45 through 48 0.5000 0.0918 + 0.3270i 0.3841 + 0.3851i    0 − 0.5000i 0.4045 − 0.4815i 0.6580 − 0.2285i −0.5000   −0.1311 − 0.6387i   −0.3448 − 0.0735i      0 − 0.5000i 0.0541 − 0.2473i 0.3076 + 0.0056i 0.5000 0.4422 + 0.0928i    0 − 0.5000i 0.2102 − 0.0137i −0.5000   −0.2479 − 0.5606i      0 − 0.5000i 0.5606 − 0.2479i columns 49 through 52 0.5000 0.3841 + 0.3851i −0.1560 + 0.4926i   0.3536 + 0.3536i 0.0022 + 0.1690i 0.0837 + 0.4175i    0 + 0.5000i −0.3076 − 0.0056i   −0.4597 + 0.3989i   −0.3536 + 0.3536i   −0.7536 − 0.1140i   −0.4175 + 0.0837i   0.5000 0.3841 + 0.3851i 0.3536 + 0.3536i −0.1140 + 0.7536i      0 + 0.5000i −0.3076 − 0.0056i   −0.3536 + 0.3536i   −0.1690 + 0.0022i   columns 53 through 56 0.5000 0.3396 + 0.1614i 0.3841 + 0.3851i −0.3536 + 0.3536i   −0.3400 − 0.3060i   −0.1690 + 0.0022i      0 − 0.5000i 0.3493 − 0.5641i 0.3076 + 0.0056i 0.3536 + 0.3536i −0.3060 + 0.3400i   −0.1140 + 0.7536i   0.5000 −0.1560 + 0.4926i   −0.3536 + 0.3536i   −0.4175 + 0.0837i      0 − 0.5000i 0.4597 − 0.3989i 0.3536 + 0.3536i 0.0837 + 0.4175i columns 57 through 60 0.5000 0.3841 + 0.3851i 0.3396 + 0.1614i −0.3536 − 0.3536i   0.1140 − 0.7536i 0.3060 − 0.3400i    0 + 0.5000i −0.3076 − 0.0056i   −0.3493 + 0.5641i   0.3536 − 0.3536i 0.1690 − 0.0022i 0.3400 + 0.3060i 0.5000 0.3841 + 0.3851i −0.3536 − 0.3536i   −0.0022 − 0.1690i      0 + 0.5000i −0.3076 − 0.0056i   0.3536 − 0.3536i 0.7536 + 0.1140i columns 61 through 64 0.5000 −0.1560 + 0.4926i   0.3841 + 0.3851i 0.3536 − 0.3536i 0.4175 − 0.0837i 0.7536 + 0.1140i    0 − 0.5000i 0.4597 − 0.3989i 0.3076 + 0.0056i −0.3536 − 0.3536i   −0.0837 − 0.4175i   −0.0022 − 0.1690i   0.5000 0.3396 + 0.1614i 0.3536 − 0.3536i 0.3400 + 0.3060i    0 − 0.5000i 0.3493 − 0.5641i −0.3536 − 0.3536i   0.3060 − 0.3400i.


2. A terminal in a multiple input multiple output (MIMO) communication system, the terminal comprising: a memory configured to store a 6-bit codebook comprising 64 codeword matrices; a channel estimator configured to estimate a channel formed between a base station and the terminal; a processor configured to select a preferred codeword vector from the codebook stored in the memory; and a feedback unit configured to feed back feedback data associated with the preferred codeword vector to the base station, wherein the 6-bit codebook comprises columns 1 through 64: columns 1 through 4 0.5000 0.3049 + 0.6229i 0.1779 + 0.0995i 0.0191 + 0.3371i 0.5000 0.2626 + 0.1626i 0.5060 + 0.2740i 0.1673 + 0.6390i 0.5000 0.4213 + 0.2108i 0.0614 + 0.5116i 0.3261 + 0.1156i 0.5000 −0.0232 + 0.4484i   0.2202 + 0.5598i 0.4531 + 0.3532i columns 5 through 8 0.5000 0.0191 + 0.3371i 0.4637 + 0.3853i 0.3049 + 0.6229i −0.5000   −0.1673 − 0.6390i   −0.0297 − 0.3692i   −0.2626 − 0.1626i   0.5000 0.3261 + 0.1156i 0.1567 + 0.6069i 0.4213 + 0.2108i −0.5000   −0.4531 − 0.3532i   −0.3155 − 0.0834i   0.0232 − 0.4484i columns 9 through 12 −0.5000   −0.1779 − 0.3853i   −0.0191 − 0.6229i   −0.4637 − 0.0995i   −0.5000   −0.2202 − 0.0834i   −0.4531 − 0.4484i   −0.3155 − 0.5598i   0.5000 0.0614 + 0.6069i 0.3261 + 0.2108i 0.1567 + 0.5116i 0.5000 0.5060 + 0.3692i 0.1673 + 0.1626i 0.0297 + 0.2740i columns 13 through 16 −0.5000   −0.4637 − 0.0995i   −0.3049 − 0.3371i   −0.1779 − 0.3853i   0.5000 0.3155 + 0.5598i −0.0232 + 0.3532i   0.2202 + 0.0834i 0.5000 0.1567 + 0.5116i 0.4213 + 0.1156i 0.0614 + 0.6069i −0.5000   −0.0297 − 0.2740i   −0.2626 − 0.6390i   −0.5060 − 0.3692i   columns 17 through 20 0.5000 0.3049 + 0.3371i 0.1779 + 0.3853i 0.0191 + 0.6229i    0 + 0.5000i −0.3532 − 0.0232i   −0.0834 + 0.2202i   −0.4484 + 0.4531i   0.5000 0.4213 + 0.1156i 0.0614 + 0.6069i 0.3261 + 0.2108i    0 + 0.5000i −0.6390 + 0.2626i   −0.3692 + 0.5060i   −0.1626 + 0.1673i   columns 21 through 24 0.5000 0.0191 + 0.6229i 0.4637 + 0.0995i 0.3049 + 0.3371i    0 − 0.5000i 0.4484 − 0.4531i 0.5598 − 0.3155i 0.3532 + 0.0232i 0.5000 0.3261 + 0.2108i 0.1567 + 0.5116i 0.4213 + 0.1156i    0 − 0.5000i 0.1626 − 0.1673i 0.2740 − 0.0297i 0.6390 − 0.2626i columns 25 through 28 −0.5000   −0.4637 − 0.3853i   −0.3049 − 0.6229i   −0.1779 − 0.0995i      0 − 0.5000i 0.3692 − 0.0297i 0.1626 − 0.2626i 0.2740 − 0.5060i 0.5000 0.1567 + 0.6069i 0.4213 + 0.2108i 0.0614 + 0.5116i    0 + 0.5000i −0.0834 + 0.3155i   −0.4484 − 0.0232i   −0.5598 + 0.2202i   columns 29 through 32 −0.5000   −0.1779 − 0.0995i   −0.0191 − 0.3371i   −0.4637 − 0.3853i      0 + 0.5000i −0.2740 + 0.5060i   −0.6390 + 0.1673i   −0.3692 + 0.0297i   0.5000 0.0614 + 0.5116i 0.3261 + 0.1156i 0.1567 + 0.6069i    0 − 0.5000i 0.5598 − 0.2202i 0.3532 − 0.4531i 0.0834 − 0.3155i columns 33 through 36 0.5000 0.0191 + 0.6229i 0.1779 + 0.3853i 0.3049 + 0.3371i 0.5000 0.1673 + 0.1626i 0.5060 + 0.3692i 0.2626 + 0.6390i 0.5000 0.3261 + 0.2108i 0.0614 + 0.6069i 0.4213 + 0.1156i −0.5000   −0.4531 − 0.4484i   −0.2202 − 0.0834i   0.0232 − 0.3532i columns 37 through 40 0.5000 0.4637 + 0.0995i 0.0191 + 0.6229i 0.1779 + 0.3853i    0 + 0.5000i −0.2740 + 0.0297i   −0.1626 + 0.1673i   −0.3692 + 0.5060i   −0.5000   −0.1567 − 0.5116i   −0.3261 − 0.2108i   −0.0614 − 0.6069i      0 + 0.5000i −0.5598 + 0.3155i   −0.4484 + 0.4531i   −0.0834 + 0.2202i   columns 41 through 44 0.5000 0.3049 + 0.3371i 0.4637 + 0.0995i 0.0191 + 0.6229i −0.5000   −0.2626 − 0.6390i   −0.0297 − 0.2740i   −0.1673 − 0.1626i   0.5000 0.4213 + 0.1156i 0.1567 + 0.5116i 0.3261 + 0.2108i 0.5000 −0.0232 + 0.3532i   0.3155 + 0.5598i 0.4531 + 0.4484i columns 45 through 48 0.5000 0.1779 + 0.3853i 0.3049 + 0.3371i 0.4637 + 0.0995i    0 − 0.5000i 0.3692 − 0.5060i 0.6390 − 0.2626i 0.2740 − 0.0297i −0.5000   −0.0614 − 0.6069i   −0.4213 − 0.1156i   −0.1567 − 0.5116i      0 − 0.5000i 0.0834 − 0.2202i 0.3532 + 0.0232i 0.5598 − 0.3155i columns 49 through 52 0.5000 0.3602 + 0.4406i −0.0795 + 0.4839i   0.3602 + 0.4406i 0.3536 + 0.3536i −0.0754 + 0.1870i   0.0965 + 0.3794i −0.0754 + 0.7586i      0 + 0.5000i −0.2289 + 0.0434i   −0.5609 + 0.3720i   −0.2289 + 0.0434i   −0.3536 + 0.3536i   −0.7586 − 0.0754i   −0.3794 + 0.0965i   −0.1870 − 0.0754i   columns 53 through 56 0.5000 0.3247 + 0.0797i 0.3602 + 0.4406i −0.0795 + 0.4839i   −0.3536 + 0.3536i   −0.3794 − 0.2846i   −0.1870 − 0.0754i   −0.3794 + 0.0965i      0 − 0.5000i 0.4262 − 0.5068i 0.2289 − 0.0434i 0.5609 − 0.3720i 0.3536 + 0.3536i −0.2846 + 0.3794i   −0.0754 + 0.7586i   0.0965 + 0.3794i columns 57 through 60 0.5000 0.3602 + 0.4406i 0.3247 + 0.0797i 0.3602 + 0.4406i −0.3536 − 0.3536i   0.0754 − 0.7586i 0.2846 − 0.3794i 0.0754 − 0.1870i    0 + 0.5000i −0.2289 + 0.0434i   −0.4262 + 0.5068i   −0.2289 + 0.0434i   0.3536 − 0.3536i 0.1870 + 0.0754i 0.3794 + 0.2846i 0.7586 + 0.0754i columns 61 through 64 0.5000 −0.0795 + 0.4839i   0.3602 + 0.4406i 0.3247 + 0.0797i 0.3536 − 0.3536i 0.3794 − 0.0965i 0.7586 + 0.0754i 0.3794 + 0.2846i    0 − 0.5000i 0.5609 − 0.3720i 0.2289 − 0.0434i 0.4262 − 0.5068i −0.3536 − 0.3536i   −0.0965 − 0.3794i   0.0754 − 0.1870i 0.2846 − 0.3794i.


3. A base station in a multiple input multiple output (MIMO) communication system, the base station comprising: a memory configured to store a 6-bit codebook comprising 64 codeword matrices; a processor configured to verify a preferred codeword vector of a terminal from the codebook stored in the memory; and a precoder configured to precode at least one data stream using a precoding matrix corresponding to the preferred codeword vector, wherein the 6-bit codebook comprises columns 1 through 64: columns 1 through 4 0.5000 0.3260 + 0.6774i 0.1499 + 0.0347i 0.0918 + 0.3270i 0.5000 0.3254 + 0.1709i 0.5009 + 0.3071i 0.1311 + 0.6387i 0.5000 0.3254 + 0.1709i 0.1505 + 0.5412i 0.2473 + 0.0541i 0.5000 −0.0250 + 0.4051i   0.1505 + 0.5412i 0.4815 + 0.4045i columns 5 through 8 0.5000 0.0918 + 0.3270i 0.3841 + 0.3851i 0.3260 + 0.6774i −0.5000   −0.1311 − 0.6387i   0.0056 − 0.3076i −0.3254 − 0.1709i   0.5000 0.2473 + 0.0541i 0.2285 + 0.6580i 0.3254 + 0.1709i −0.5000   −0.4815 − 0.4045i   −0.3448 − 0.0735i   0.0250 − 0.4051i columns 9 through 12 −0.5000   −0.0918 − 0.3270i   −0.0337 − 0.6193i   −0.4422 − 0.0928i   −0.5000   −0.2473 − 0.0541i   −0.4621 − 0.5019i   −0.2479 − 0.5606i   0.5000 0.1311 + 0.6387i 0.2280 + 0.1515i 0.2479 + 0.5606i 0.5000 0.4815 + 0.4045i 0.2280 + 0.1515i 0.0137 + 0.2102i columns 13 through 16 −0.5000   −0.4422 − 0.0928i   −0.3841 − 0.3851i   −0.0918 − 0.3270i   0.5000 0.2479 + 0.5606i −0.0056 + 0.3076i   0.2473 + 0.0541i 0.5000 0.2479 + 0.5606i 0.3448 + 0.0735i 0.1311 + 0.6387i −0.5000   −0.0137 − 0.2102i   −0.2285 − 0.6580i   −0.4815 − 0.4045i   columns 17 through 20 0.5000 0.3841 + 0.3851i 0.0918 + 0.3270i 0.0337 + 0.6193i    0 + 0.5000i −0.3076 − 0.0056i   −0.0541 + 0.2473i   −0.5019 + 0.4621i   0.5000 0.3448 + 0.0735i 0.1311 + 0.6387i 0.2280 + 0.1515i    0 + 0.5000i −0.6580 + 0.2285i   −0.4045 + 0.4815i   −0.1515 + 0.2280i   columns 21 through 24 0.5000 0.0337 + 0.6193i 0.4422 + 0.0928i 0.3841 + 0.3851i    0 − 0.5000i 0.5019 − 0.4621i 0.5606 − 0.2479i 0.3076 + 0.0056i 0.5000 0.2280 + 0.1515i 0.2479 + 0.5606i 0.3448 + 0.0735i    0 − 0.5000i 0.1515 − 0.2280i 0.2102 − 0.0137i 0.6580 − 0.2285i columns 25 through 28 −0.5000   −0.3841 − 0.3851i   − 0.3260 − 0.6774i   −0.1499 − 0.0347i      0 − 0.5000i 0.3076 + 0.0056i 0.1709 − 0.3254i 0.3071 − 0.5009i 0.5000 0.2285 + 0.6580i 0.3254 + 0.1709i 0.1505 + 0.5412i    0 + 0.5000i −0.0735 + 0.3448i   −0.4051 − 0.0250i   −0.5412 + 0.1505i   columns 29 through 32 −0.5000   −0.1499 − 0.0347i   −0.0918 − 0.3270i   −0.3841 − 0.3851i      0 + 0.5000i −0.3071 + 0.5009i   −0.6387 + 0.1311i   −0.3076 − 0.0056i   0.5000 0.1505 + 0.5412i 0.2473 + 0.0541i 0.2285 + 0.6580i    0 − 0.5000i 0.5412 − 0.1505i 0.4045 − 0.4815i 0.0735 − 0.3448i columns 33 through 36 0.5000 0.0337 + 0.6193i 0.0918 + 0.3270i 0.3841 + 0.3851i 0.5000 0.2280 + 0.1515i 0.4815 + 0.4045i 0.2285 + 0.6580i 0.5000 0.2280 + 0.1515i 0.1311 + 0.6387i 0.3448 + 0.0735i −0.5000   −0.4621 − 0.5019i   −0.2473 − 0.0541i   0.0056 − 0.3076i columns 37 through 40 0.5000 0.4422 + 0.0928i 0.0337 + 0.6193i 0.0918 + 0.3270i    0 + 0.5000i −0.2102 + 0.0137i   −0.1515 + 0.2280i   −0.4045 + 0.4815i   −0.5000   −0.2479 − 0.5606i   −0.2280 − 0.1515i   −0.1311 − 0.6387i      0 + 0.5000i −0.5606 + 0.2479i   −0.5019 + 0.4621i   −0.0541 + 0.2473i   columns 41 through 44 0.5000 0.3841 + 0.3851i 0.4422 + 0.0928i 0.0337 + 0.6193i −0.5000   −0.2285 − 0.6580i   −0.0137 − 0.2102i   −0.2280 − 0.1515i   0.5000 0.3448 + 0.0735i 0.2479 + 0.5606i 0.2280 + 0.1515i 0.5000 −0.0056 + 0.3076i   0.2479 + 0.5606i 0.4621 + 0.5019i columns 45 through 48 0.5000 0.0918 + 0.3270i 0.3841 + 0.3851i 0.4422 + 0.0928i    0 − 0.5000i 0.4045 − 0.4815i 0.6580 − 0.2285i 0.2102 − 0.0137i −0.5000   −0.1311 − 0.6387i   −0.3448 − 0.0735i   −0.2479 − 0.5606i      0 − 0.5000i 0.0541 − 0.2473i 0.3076 + 0.0056i 0.5606 − 0.2479i columns 49 through 52 0.5000 0.3841 + 0.3851i −0.1560 + 0.4926i   0.3841 + 0.3851i 0.3536 + 0.3536i 0.0022 + 0.1690i 0.0837 + 0.4175i −0.1140 + 0.7536i   0 + 0.5000i −0.3076 − 0.0056i   −0.4597 + 0.3989i   −0.3076 − 0.0056i   −0.3536 + 0.3536i   −0.7536 − 0.1140i   −0.4175 + 0.0837i   −0.1690 + 0.0022i   columns 53 through 56 0.5000 0.3396 + 0.1614i 0.3841 + 0.3851i −0.1560 + 0.4926i   −0.3536 + 0.3536i −0.3400 − 0.3060i   −0.1690 + 0.0022i   −0.4175 + 0.0837i      0 − 0.5000i 0.3493 − 0.5641i 0.3076 + 0.0056i 0.4597 − 0.3989i 0.3536 + 0.3536i −0.3060 + 0.3400i   −0.1140 + 0.7536i   0.0837 + 0.4175i columns 57 through 60 0.5000 0.3841 + 0.3851i 0.3396 + 0.1614i 0.3841 + 0.3851i −0.3536 − 0.3536i   0.1140 − 0.7536i 0.3060 − 0.3400i −0.0022 − 0.1690i      0 + 0.5000i −0.3076 − 0.0056i   −0.3493 + 0.5641i   −0.3076 − 0.0056i   0.3536 − 0.3536i 0.1690 − 0.0022i 0.3400 + 0.3060i 0.7536 + 0.1140i columns 61 through 64 0.5000 −0.1560 + 0.4926i   0.3841 + 0.3851i 0.3396 + 0.1614i 0.3536 − 0.3536i 0.4175 − 0.0837i 0.7536 + 0.1140i 0.3400 + 0.3060i    0 − 0.5000i 0.4597 − 0.3989i 0.3076 + 0.0056i 0.3493 − 0.5641i −0.3536 − 0.3536i   −0.0837 − 0.4175i   −0.0022 − 0.1690i   0.3060 − 0.3400i.


4. A base station in a multiple input multiple output (MIMO) communication system, the base station comprising: a memory configured to store a 6-bit codebook comprising 64 codeword matrices; a processor configured to verify a preferred codeword vector of a terminal from the codebook stored in the memory; and a precoder configured to precode at least one data stream using a precoding matrix corresponding to the preferred codeword vector, wherein the 6-bit codebook comprises columns 1 through 64: columns 1 through 4 0.5000 0.3049 + 0.6229i 0.1779 + 0.0995i 0.0191 + 0.3371i 0.5000 0.2626 + 0.1626i 0.5060 + 0.2740i 0.1673 + 0.6390i 0.5000 0.4213 + 0.2108i 0.0614 + 0.5116i 0.3261 + 0.1156i 0.5000 −0.0232 + 0.4484i   0.2202 + 0.5598i 0.4531 + 0.3532i columns 5 through 8 0.5000 0.0191 + 0.3371i 0.4637 + 0.3853i 0.3049 + 0.6229i −0.5000   −0.1673 − 0.6390i   −0.0297 − 0.3692i   −0.2626 − 0.1626i   0.5000 0.3261 + 0.1156i 0.1567 + 0.6069i 0.4213 + 0.2108i −0.5000   −0.4531 − 0.3532i   −0.3155 − 0.0834i   0.0232 − 0.4484i columns 9 through 12 −0.5000   −0.1779 − 0.3853i   −0.0191 − 0.6229i   −0.4637 − 0.0995i   −0.5000   −0.2202 − 0.0834i   −0.4531 − 0.4484i   −0.3155 − 0.5598i   0.5000 0.0614 + 0.6069i 0.3261 + 0.2108i 0.1567 + 0.5116i 0.5000 0.5060 + 0.3692i 0.1673 + 0.1626i 0.0297 + 0.2740i columns 13 through 16 −0.5000   −0.4637 − 0.0995i   −0.3049 − 0.3371i   −0.1779 − 0.3853i   0.5000 0.3155 + 0.5598i −0.0232 + 0.3532i   0.2202 + 0.0834i 0.5000 0.1567 + 0.5116i 0.4213 + 0.1156i 0.0614 + 0.6069i −0.5000   −0.0297 − 0.2740i   −0.2626 − 0.6390i   −0.5060 − 0.3692i   columns 17 through 20 0.5000 0.3049 + 0.3371i 0.1779 + 0.3853i 0.0191 + 0.6229i    0 + 0.5000i −0.3532 − 0.0232i   −0.0834 + 0.2202i   −0.4484 + 0.4531i   0.5000 0.4213 + 0.1156i 0.0614 + 0.6069i 0.3261 + 0.2108i    0 + 0.5000i −0.6390 + 0.2626i   −0.3692 + 0.5060i   −0.1626 + 0.1673i   columns 21 through 24 0.5000 0.0191 + 0.6229i 0.4637 + 0.0995i 0.3049 + 0.3371i    0 − 0.5000i 0.4484 − 0.4531i 0.5598 − 0.3155i 0.3532 + 0.0232i 0.5000 0.3261 + 0.2108i 0.1567 + 0.5116i 0.4213 + 0.1156i    0 − 0.5000i 0.1626 − 0.1673i 0.2740 − 0.0297i 0.6390 − 0.2626i columns 25 through 28 −0.5000   −0.4637 − 0.3853i   −0.3049 − 0.6229i   −0.1779 − 0.0995i      0 − 0.5000i 0.3692 − 0.0297i 0.1626 − 0.2626i 0.2740 − 0.5060i 0.5000 0.1567 + 0.6069i 0.4213 + 0.2108i 0.0614 + 0.5116i    0 + 0.5000i −0.0834 + 0.3155i   −0.4484 − 0.0232i   −0.5598 + 0.2202i   columns 29 through 32 −0.5000   −0.1779 − 0.0995i   −0.0191 − 0.3371i   −0.4637 − 0.3853i      0 + 0.5000i −0.2740 + 0.5060i   −0.6390 + 0.1673i   −0.3692 + 0.0297i   0.5000 0.0614 + 0.5116i 0.3261 + 0.1156i 0.1567 + 0.6069i    0 − 0.5000i 0.5598 − 0.2202i 0.3532 − 0.4531i 0.0834 − 0.3155i columns 33 through 36 0.5000 0.0191 + 0.6229i 0.1779 + 0.3853i 0.3049 + 0.3371i 0.5000 0.1673 + 0.1626i 0.5060 + 0.3692i 0.2626 + 0.6390i 0.5000 0.3261 + 0.2108i 0.0614 + 0.6069i 0.4213 + 0.1156i −0.5000   −0.4531 − 0.4484i   −0.2202 − 0.0834i   0.0232 − 0.3532i columns 37 through 40 0.5000 0.4637 + 0.0995i 0.0191 + 0.6229i 0.1779 + 0.3853i    0 + 0.5000i −0.2740 + 0.0297i   −0.1626 + 0.1673i   −0.3692 + 0.5060i   −0.5000   −0.1567 − 0.5116i   −0.3261 − 0.2108i   −0.0614 − 0.6069i      0 + 0.5000i −0.5598 + 0.3155i   −0.4484 + 0.4531i   −0.0834 + 0.2202i   columns 41 through 44 0.5000 0.3049 + 0.3371i 0.4637 + 0.0995i 0.0191 + 0.6229i −0.5000   −0.2626 − 0.6390i   −0.0297 − 0.2740i   −0.1673 − 0.1626i   0.5000 0.4213 + 0.1156i 0.1567 + 0.5116i 0.3261 + 0.2108i 0.5000 −0.0232 + 0.3532i   0.3155 + 0.5598i 0.4531 + 0.4484i columns 45 through 48 0.5000 0.1779 + 0.3853i 0.3049 + 0.3371i 0.4637 + 0.0995i    0 − 0.5000i 0.3692 − 0.5060i 0.6390 − 0.2626i 0.2740 − 0.0297i −0.5000   −0.0614 − 0.6069i   −0.4213 − 0.1156i   −0.1567 − 0.5116i      0 − 0.5000i 0.0834 − 0.2202i 0.3532 + 0.0232i 0.5598 − 0.3155i columns 49 through 52 0.5000 0.3602 + 0.4406i −0.0795 + 0.4839i   0.3602 + 0.4406i 0.3536 + 0.3536i −0.0754 + 0.1870i   0.0965 + 0.3794i −0.0754 + 0.7586i      0 + 0.5000i −0.2289 + 0.0434i   −0.5609 + 0.3720i   −0.2289 + 0.0434i   −0.3536 + 0.3536i   −0.7586 − 0.0754i   −0.3794 + 0.0965i   −0.1870 − 0.0754i   columns 53 through 56 0.5000 0.3247 + 0.0797i 0.3602 + 0.4406i −0.0795 + 0.4839i   −0.3536 + 0.3536i   −0.3794 − 0.2846i   −0.1870 − 0.0754i   −0.3794 + 0.0965i      0 − 0.5000i 0.4262 − 0.5068i 0.2289 − 0.0434i 0.5609 − 0.3720i 0.3536 + 0.3536i −0.2846 + 0.3794i   −0.0754 + 0.7586i   0.0965 + 0.3794i columns 57 through 60 0.5000 0.3602 + 0.4406i 0.3247 + 0.0797i 0.3602 + 0.4406i −0.3536 − 0.3536i   0.0754 − 0.7586i 0.2846 − 0.3794i 0.0754 − 0.1870i    0 + 0.5000i −0.2289 + 0.0434i   −0.4262 + 0.5068i   −0.2289 + 0.0434i   0.3536 − 0.3536i 0.1870 + 0.0754i 0.3794 + 0.2846i 0.7586 + 0.0754i columns 61 through 64 0.5000 −0.0795 + 0.4839i   0.3602 + 0.4406i 0.3247 + 0.0797i 0.3536 − 0.3536i 0.3794 − 0.0965i 0.7586 + 0.0754i 0.3794 + 0.2846i    0 − 0.5000i 0.5609 − 0.3720i 0.2289 − 0.0434i 0.4262 − 0.5068i −0.3536 − 0.3536i   −0.0965 − 0.3794i   0.0754 − 0.1870i  0.2846 − 0.3794i. 